diff --git a/Library/Homebrew/cmd/--prefix.rb b/Library/Homebrew/cmd/--prefix.rb
index c01fa7dc17..2f1a87fe80 100644
--- a/Library/Homebrew/cmd/--prefix.rb
+++ b/Library/Homebrew/cmd/--prefix.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "abstract_command"
@@ -91,6 +91,7 @@ module Homebrew
 
       private
 
+      sig { void }
       def list_unbrewed
         dirs  = HOMEBREW_PREFIX.subdirs.map { |dir| dir.basename.to_s }
         dirs -= %w[Library Cellar Caskroom .git]
diff --git a/Library/Homebrew/cmd/gist-logs.rb b/Library/Homebrew/cmd/gist-logs.rb
index 8ab830d8ca..dd3b5e71b0 100644
--- a/Library/Homebrew/cmd/gist-logs.rb
+++ b/Library/Homebrew/cmd/gist-logs.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "abstract_command"
@@ -33,11 +33,14 @@ module Homebrew
       def run
         Install.perform_preinstall_checks(all_fatal: true)
         Install.perform_build_from_source_checks(all_fatal: true)
-        gistify_logs(args.named.to_resolved_formulae.first)
+        return unless (formula = args.named.to_resolved_formulae.first)
+
+        gistify_logs(formula)
       end
 
       private
 
+      sig { params(formula: Formula).void }
       def gistify_logs(formula)
         files = load_logs(formula.logs)
         build_time = formula.logs.ctime
@@ -86,6 +89,7 @@ module Homebrew
         puts url if url
       end
 
+      sig { params(formula: Formula, with_hostname: T::Boolean).returns(String) }
       def brief_build_info(formula, with_hostname:)
         build_time_string = formula.logs.ctime.strftime("%Y-%m-%d %H:%M:%S")
         string = +<<~EOS
@@ -100,6 +104,7 @@ module Homebrew
       end
 
       # Causes some terminals to display secure password entry indicators.
+      sig { void }
       def noecho_gets
         system "stty", "-echo"
         result = $stdin.gets
@@ -108,6 +113,7 @@ module Homebrew
         result
       end
 
+      sig { params(dir: Pathname, basedir: Pathname).returns(T::Hash[String, T::Hash[Symbol, String]]) }
       def load_logs(dir, basedir = dir)
         logs = {}
         if dir.exist?
diff --git a/Library/Homebrew/cmd/home.rb b/Library/Homebrew/cmd/home.rb
index b10ccb406a..8def8448c5 100644
--- a/Library/Homebrew/cmd/home.rb
+++ b/Library/Homebrew/cmd/home.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "abstract_command"
@@ -41,6 +41,7 @@ module Homebrew
 
       private
 
+      sig { params(formula_or_cask: T.any(Formula, Cask::Cask)).returns(String) }
       def name_of(formula_or_cask)
         if formula_or_cask.is_a? Formula
           "Formula #{formula_or_cask.name}"
diff --git a/Library/Homebrew/cmd/leaves.rb b/Library/Homebrew/cmd/leaves.rb
index b8b0061505..8bf66f3f46 100644
--- a/Library/Homebrew/cmd/leaves.rb
+++ b/Library/Homebrew/cmd/leaves.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "abstract_command"
@@ -39,12 +39,14 @@ module Homebrew
 
       private
 
+      sig { params(formula: Formula).returns(T::Boolean) }
       def installed_on_request?(formula)
-        formula.any_installed_keg.tab.installed_on_request
+        formula.any_installed_keg&.tab&.installed_on_request
       end
 
+      sig { params(formula: Formula).returns(T::Boolean) }
       def installed_as_dependency?(formula)
-        formula.any_installed_keg.tab.installed_as_dependency
+        formula.any_installed_keg&.tab&.installed_as_dependency
       end
     end
   end
diff --git a/Library/Homebrew/cmd/link.rb b/Library/Homebrew/cmd/link.rb
index 9c43e48e96..c185106751 100644
--- a/Library/Homebrew/cmd/link.rb
+++ b/Library/Homebrew/cmd/link.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "abstract_command"
@@ -123,6 +123,7 @@ module Homebrew
 
       private
 
+      sig { params(keg: Keg).void }
       def puts_keg_only_path_message(keg)
         bin = keg/"bin"
         sbin = keg/"sbin"
diff --git a/Library/Homebrew/cmd/list.rb b/Library/Homebrew/cmd/list.rb
index 7652cbe462..509b91a166 100644
--- a/Library/Homebrew/cmd/list.rb
+++ b/Library/Homebrew/cmd/list.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "abstract_command"
@@ -160,6 +160,7 @@ module Homebrew
 
       private
 
+      sig { void }
       def filtered_list
         names = if args.no_named?
           Formula.racks
@@ -189,6 +190,7 @@ module Homebrew
         end
       end
 
+      sig { void }
       def list_casks
         casks = if args.no_named?
           Cask::Caskroom.casks
@@ -212,6 +214,7 @@ module Homebrew
     end
 
     class PrettyListing
+      sig { params(path: T.any(String, Pathname, Keg)).void }
       def initialize(path)
         Pathname.new(path).children.sort_by { |p| p.to_s.downcase }.each do |pn|
           case pn.basename.to_s
@@ -240,7 +243,8 @@ module Homebrew
 
       private
 
-      def print_dir(root)
+      sig { params(root: Pathname, block: T.nilable(T.proc.params(arg0: Pathname).returns(T::Boolean))).void }
+      def print_dir(root, &block)
         dirs = []
         remaining_root_files = []
         other = ""
@@ -248,7 +252,7 @@ module Homebrew
         root.children.sort.each do |pn|
           if pn.directory?
             dirs << pn
-          elsif block_given? && yield(pn)
+          elsif block && yield(pn)
             puts pn
             other = "other "
           elsif pn.basename.to_s != ".DS_Store"
@@ -265,6 +269,7 @@ module Homebrew
         print_remaining_files remaining_root_files, root, other
       end
 
+      sig { params(files: T::Array[Pathname], root: Pathname, other: String).void }
       def print_remaining_files(files, root, other = "")
         if files.length == 1
           puts files
diff --git a/Library/Homebrew/cmd/options.rb b/Library/Homebrew/cmd/options.rb
index f316890145..ad62093461 100644
--- a/Library/Homebrew/cmd/options.rb
+++ b/Library/Homebrew/cmd/options.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "abstract_command"
@@ -54,6 +54,7 @@ module Homebrew
 
       private
 
+      sig { params(formulae: T::Array[Formula]).void }
       def puts_options(formulae)
         formulae.each do |f|
           next if f.options.empty?
diff --git a/Library/Homebrew/cmd/outdated.rb b/Library/Homebrew/cmd/outdated.rb
index 710e282c4a..17232e8dd4 100644
--- a/Library/Homebrew/cmd/outdated.rb
+++ b/Library/Homebrew/cmd/outdated.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "abstract_command"
@@ -86,6 +86,7 @@ module Homebrew
 
       private
 
+      sig { params(formulae_or_casks: T::Array[T.any(Formula, Cask::Cask)]).void }
       def print_outdated(formulae_or_casks)
         formulae_or_casks.each do |formula_or_cask|
           if formula_or_cask.is_a?(Formula)
@@ -124,6 +125,13 @@ module Homebrew
         end
       end
 
+      sig {
+        params(
+          formulae_or_casks: T::Array[T.any(Formula, Cask::Cask)],
+        ).returns(
+          T::Array[T.any(T::Hash[String, T.untyped], T::Hash[String, T.untyped])],
+        )
+      }
       def json_info(formulae_or_casks)
         formulae_or_casks.map do |formula_or_cask|
           if formula_or_cask.is_a?(Formula)
@@ -149,10 +157,12 @@ module Homebrew
         end
       end
 
+      sig { returns(T::Boolean) }
       def verbose?
         ($stdout.tty? || Context.current.verbose?) && !Context.current.quiet?
       end
 
+      sig { params(version: T.nilable(T.any(TrueClass, String))).returns(T.nilable(Symbol)) }
       def json_version(version)
         version_hash = {
           nil  => nil,
@@ -166,18 +176,26 @@ module Homebrew
         version_hash[version]
       end
 
+      sig { returns(T::Array[Formula]) }
       def outdated_formulae
-        select_outdated((args.named.to_resolved_formulae.presence || Formula.installed)).sort
+        T.cast(
+          select_outdated((args.named.to_resolved_formulae.presence || Formula.installed)).sort,
+          T::Array[Formula],
+        )
       end
 
+      sig { returns(T::Array[Cask::Cask]) }
       def outdated_casks
-        if args.named.present?
+        outdated = if args.named.present?
           select_outdated(args.named.to_casks)
         else
           select_outdated(Cask::Caskroom.casks)
         end
+
+        T.cast(outdated, T::Array[Cask::Cask])
       end
 
+      sig { returns([T::Array[T.any(Formula, Cask::Cask)], T::Array[T.any(Formula, Cask::Cask)]]) }
       def outdated_formulae_casks
         formulae, casks = args.named.to_resolved_formulae_to_casks
 
@@ -189,6 +207,9 @@ module Homebrew
         [select_outdated(formulae).sort, select_outdated(casks)]
       end
 
+      sig {
+        params(formulae_or_casks: T::Array[T.any(Formula, Cask::Cask)]).returns(T::Array[T.any(Formula, Cask::Cask)])
+      }
       def select_outdated(formulae_or_casks)
         formulae_or_casks.select do |formula_or_cask|
           if formula_or_cask.is_a?(Formula)
diff --git a/Library/Homebrew/cmd/search.rb b/Library/Homebrew/cmd/search.rb
index 8c6d89a344..9d0e6cf6bd 100644
--- a/Library/Homebrew/cmd/search.rb
+++ b/Library/Homebrew/cmd/search.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "abstract_command"
@@ -10,7 +10,7 @@ require "search"
 module Homebrew
   module Cmd
     class SearchCmd < AbstractCommand
-      PACKAGE_MANAGERS = {
+      PACKAGE_MANAGERS = T.let({
         repology:  ->(query) { "https://repology.org/projects/?search=#{query}" },
         macports:  ->(query) { "https://ports.macports.org/search/?q=#{query}" },
         fink:      ->(query) { "https://pdb.finkproject.org/pdb/browse.php?summary=#{query}" },
@@ -23,7 +23,7 @@ module Homebrew
         ubuntu:    lambda { |query|
           "https://packages.ubuntu.com/search?keywords=#{query}&searchon=names&suite=all&section=all"
         },
-      }.freeze
+      }.freeze, T::Hash[Symbol, T.proc.params(query: String).returns(String)])
 
       cmd_args do
         description <<~EOS
@@ -89,6 +89,7 @@ module Homebrew
 
       private
 
+      sig { void }
       def print_regex_help
         return unless $stdout.tty?
 
@@ -105,6 +106,7 @@ module Homebrew
         EOS
       end
 
+      sig { returns(T::Boolean) }
       def search_package_manager
         package_manager = PACKAGE_MANAGERS.find { |name,| args[:"#{name}?"] }
         return false if package_manager.nil?
@@ -114,6 +116,7 @@ module Homebrew
         true
       end
 
+      sig { params(query: String).returns(String) }
       def search_pull_requests(query)
         only = if args.open? && !args.closed?
           "open"
@@ -124,6 +127,7 @@ module Homebrew
         GitHub.print_pull_requests_matching(query, only)
       end
 
+      sig { params(all_formulae: T::Array[String], all_casks: T::Array[String], query: String).void }
       def print_results(all_formulae, all_casks, query)
         count = all_formulae.size + all_casks.size
 
@@ -148,6 +152,7 @@ module Homebrew
         odie "No formulae or casks found for #{query.inspect}." if count.zero?
       end
 
+      sig { params(query: String, found_matches: T::Boolean).void }
       def print_missing_formula_help(query, found_matches)
         return unless $stdout.tty?
 
diff --git a/Library/Homebrew/cmd/tap-info.rb b/Library/Homebrew/cmd/tap-info.rb
index f430de8ce0..2d00a70cb8 100644
--- a/Library/Homebrew/cmd/tap-info.rb
+++ b/Library/Homebrew/cmd/tap-info.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "abstract_command"
@@ -42,6 +42,7 @@ module Homebrew
 
       private
 
+      sig { params(taps: T::Array[Tap]).void }
       def print_tap_info(taps)
         if taps.none?
           tap_count = 0
@@ -83,6 +84,7 @@ module Homebrew
         end
       end
 
+      sig { params(taps: T::Array[Tap]).void }
       def print_tap_json(taps)
         puts JSON.pretty_generate(taps.map(&:to_hash))
       end
diff --git a/Library/Homebrew/cmd/uses.rb b/Library/Homebrew/cmd/uses.rb
index eea29f5e7e..3bcb86a2e6 100644
--- a/Library/Homebrew/cmd/uses.rb
+++ b/Library/Homebrew/cmd/uses.rb
@@ -1,4 +1,4 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 require "abstract_command"
@@ -87,6 +87,7 @@ module Homebrew
 
       private
 
+      sig { params(use_runtime_dependents: T::Boolean, used_formulae: T::Array[Formula]).returns(T::Array[Formula]) }
       def intersection_of_dependents(use_runtime_dependents, used_formulae)
         recursive = args.recursive?
         show_formulae_and_casks = !args.formula? && !args.cask?
@@ -137,6 +138,14 @@ module Homebrew
         end
       end
 
+      sig {
+        params(
+          dependents: T::Array[Formula], used_formulae: T::Array[Formula], recursive: T::Boolean,
+          includes: T::Array[Symbol], ignores: T::Array[Symbol]
+        ).returns(
+          T::Array[Formula],
+        )
+      }
       def select_used_dependents(dependents, used_formulae, recursive, includes, ignores)
         dependents.select do |d|
           deps = if recursive
diff --git a/Library/Homebrew/extend/os/linux/cmd/update-report.rb b/Library/Homebrew/extend/os/linux/cmd/update-report.rb
index 2d448f0ec1..25947f07c4 100644
--- a/Library/Homebrew/extend/os/linux/cmd/update-report.rb
+++ b/Library/Homebrew/extend/os/linux/cmd/update-report.rb
@@ -1,13 +1,15 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 module Homebrew
   module_function
 
+  sig { returns(String) }
   def no_changes_message
     "No changes to formulae."
   end
 
+  sig { void }
   def migrate_gcc_dependents_if_needed
     return if Settings.read("gcc-rpaths.fixed") == "true"