diff --git a/Library/.rubocop.yml b/Library/.rubocop.yml
index 3a453e8c50..83165428c6 100644
--- a/Library/.rubocop.yml
+++ b/Library/.rubocop.yml
@@ -279,6 +279,7 @@ Sorbet/FalseSigil:
     - 'Homebrew/test/**/Casks/**/*.rb'
 
 Sorbet/StrictSigil:
+  Enabled: true
   Include:
     - '**/*.rbi'
 
diff --git a/Library/Homebrew/PATH.rb b/Library/Homebrew/PATH.rb
index f9c347d6e8..20b83d0d36 100644
--- a/Library/Homebrew/PATH.rb
+++ b/Library/Homebrew/PATH.rb
@@ -1,52 +1,71 @@
-# typed: false
+# typed: true
 # frozen_string_literal: true
 
 # Represention of a `*PATH` environment variable.
 #
 # @api private
 class PATH
+  extend T::Sig
+
   include Enumerable
   extend Forwardable
 
   def_delegator :@paths, :each
 
+  # FIXME: Enable cop again when https://github.com/sorbet/sorbet/issues/3532 is fixed.
+  # rubocop:disable Style/MutableConstant
+  Element = T.type_alias { T.nilable(T.any(Pathname, String, PATH)) }
+  private_constant :Element
+  Elements = T.type_alias { T.any(Element, T::Array[Element]) }
+  private_constant :Elements
+  # rubocop:enable Style/MutableConstant
+
+  sig { params(paths: Elements).void }
   def initialize(*paths)
-    @paths = parse(*paths)
+    @paths = parse(paths)
   end
 
+  sig { params(paths: Elements).returns(T.self_type) }
   def prepend(*paths)
-    @paths = parse(*paths, *@paths)
+    @paths = parse(paths + @paths)
     self
   end
 
+  sig { params(paths: Elements).returns(T.self_type) }
   def append(*paths)
-    @paths = parse(*@paths, *paths)
+    @paths = parse(@paths + paths)
     self
   end
 
+  sig { params(index: Integer, paths: Elements).returns(T.self_type) }
   def insert(index, *paths)
-    @paths = parse(*@paths.insert(index, *paths))
+    @paths = parse(@paths.insert(index, *paths))
     self
   end
 
+  sig { params(block: T.proc.params(arg0: String).returns(T::Boolean)).returns(T.self_type) }
   def select(&block)
     self.class.new(@paths.select(&block))
   end
 
+  sig { params(block: T.proc.params(arg0: String).returns(T::Boolean)).returns(T.self_type) }
   def reject(&block)
     self.class.new(@paths.reject(&block))
   end
 
+  sig { returns(T::Array[String]) }
   def to_ary
     @paths.dup.to_ary
   end
   alias to_a to_ary
 
+  sig { returns(String) }
   def to_str
     @paths.join(File::PATH_SEPARATOR)
   end
   alias to_s to_str
 
+  sig { params(other: T.untyped).returns(T::Boolean) }
   def ==(other)
     if other.respond_to?(:to_ary) && to_ary == other.to_ary
       true
@@ -57,10 +76,12 @@ class PATH
     end
   end
 
+  sig { returns(T::Boolean) }
   def empty?
     @paths.empty?
   end
 
+  sig { returns(T.nilable(T.self_type)) }
   def existing
     existing_path = select(&File.method(:directory?))
     # return nil instead of empty PATH, to unset environment variables
@@ -69,10 +90,11 @@ class PATH
 
   private
 
-  def parse(*paths)
+  sig { params(paths: T::Array[Elements]).returns(T::Array[String]) }
+  def parse(paths)
     paths.flatten
          .compact
-         .flat_map { |p| Pathname.new(p).to_path.split(File::PATH_SEPARATOR) }
+         .flat_map { |p| Pathname(p).to_path.split(File::PATH_SEPARATOR) }
          .uniq
   end
 end
diff --git a/Library/Homebrew/extend/os/mac/unpack_strategy/zip.rb b/Library/Homebrew/extend/os/mac/unpack_strategy/zip.rb
index d90cdb21f5..b804e49f35 100644
--- a/Library/Homebrew/extend/os/mac/unpack_strategy/zip.rb
+++ b/Library/Homebrew/extend/os/mac/unpack_strategy/zip.rb
@@ -1,9 +1,19 @@
-# typed: false
+# typed: strict
 # frozen_string_literal: true
 
+require "system_command"
+
 module UnpackStrategy
   class Zip
     module MacOSZipExtension
+      extend T::Sig
+
+      include UnpackStrategy
+      include SystemCommand::Mixin
+
+      using Magic
+
+      sig { override.params(unpack_dir: Pathname, basename: Pathname, verbose: T::Boolean).returns(T.untyped) }
       def extract_to_dir(unpack_dir, basename:, verbose:)
         if merge_xattrs && contains_extended_attributes?(path)
           # We use ditto directly, because dot_clean has issues if the __MACOSX
@@ -16,7 +26,7 @@ module UnpackStrategy
         end
 
         result = begin
-          super
+          T.let(super, SystemCommand::Result)
         rescue ErrorDuringExecution => e
           raise unless e.stderr.include?("End-of-central-directory signature not found.")
 
@@ -47,6 +57,13 @@ module UnpackStrategy
           end
         end
       end
+
+      private
+
+      sig { params(path: Pathname).returns(T::Boolean) }
+      def contains_extended_attributes?(path)
+        path.zipinfo.grep(/(^__MACOSX|\._)/).any?
+      end
     end
     private_constant :MacOSZipExtension
 
diff --git a/Library/Homebrew/extend/os/mac/unpack_strategy/zip.rbi b/Library/Homebrew/extend/os/mac/unpack_strategy/zip.rbi
new file mode 100644
index 0000000000..ee40416382
--- /dev/null
+++ b/Library/Homebrew/extend/os/mac/unpack_strategy/zip.rbi
@@ -0,0 +1,9 @@
+# typed: strict
+
+module UnpackStrategy
+  class Zip
+    module MacOSZipExtension
+      include Kernel
+    end
+  end
+end
diff --git a/Library/Homebrew/sorbet/config b/Library/Homebrew/sorbet/config
index 02fbf5ab02..024ba458ec 100644
--- a/Library/Homebrew/sorbet/config
+++ b/Library/Homebrew/sorbet/config
@@ -6,3 +6,6 @@
 
 --ignore
 /test/.gem
+
+--dsl-plugins
+sorbet/triggers.yml
diff --git a/Library/Homebrew/sorbet/plugins/unpack_strategy_magic.rb b/Library/Homebrew/sorbet/plugins/unpack_strategy_magic.rb
new file mode 100644
index 0000000000..d5c83fa8b5
--- /dev/null
+++ b/Library/Homebrew/sorbet/plugins/unpack_strategy_magic.rb
@@ -0,0 +1,21 @@
+# typed: strict
+# frozen_string_literal: true
+
+source = ARGV[5]
+
+/\busing +Magic\b/.match(source) do |_|
+  puts <<-RUBY
+    # typed: strict
+
+    class ::Pathname
+      sig { returns(String) }
+      def magic_number; end
+
+      sig { returns(String) }
+      def file_type; end
+
+      sig { returns(T::Array[String]) }
+      def zipinfo; end
+    end
+  RUBY
+end
diff --git a/Library/Homebrew/sorbet/triggers.yml b/Library/Homebrew/sorbet/triggers.yml
new file mode 100644
index 0000000000..52bc3b2aea
--- /dev/null
+++ b/Library/Homebrew/sorbet/triggers.yml
@@ -0,0 +1,5 @@
+ruby_extra_args:
+  - --disable-gems
+
+triggers:
+  using: sorbet/plugins/unpack_strategy_magic.rb
diff --git a/Library/Homebrew/unpack_strategy.rb b/Library/Homebrew/unpack_strategy.rb
index 774d296156..c187be8286 100644
--- a/Library/Homebrew/unpack_strategy.rb
+++ b/Library/Homebrew/unpack_strategy.rb
@@ -7,12 +7,16 @@ require "system_command"
 #
 # @api private
 module UnpackStrategy
+  extend T::Sig
+  extend T::Helpers
+
   include SystemCommand::Mixin
 
   # Helper module for identifying the file type.
   module Magic
     # Length of the longest regex (currently Tar).
     MAX_MAGIC_NUMBER_LENGTH = 262
+    private_constant :MAX_MAGIC_NUMBER_LENGTH
 
     refine Pathname do
       def magic_number
@@ -125,11 +129,16 @@ module UnpackStrategy
     @merge_xattrs = merge_xattrs
   end
 
-  def extract(to: nil, basename: nil, verbose: false)
+  abstract!
+  sig { abstract.params(unpack_dir: Pathname, basename: Pathname, verbose: T::Boolean).returns(T.untyped) }
+  def extract_to_dir(unpack_dir, basename:, verbose:); end
+  private :extract_to_dir
+
+  def extract(to: nil, basename: nil, verbose: nil)
     basename ||= path.basename
     unpack_dir = Pathname(to || Dir.pwd).expand_path
     unpack_dir.mkpath
-    extract_to_dir(unpack_dir, basename: basename, verbose: verbose)
+    extract_to_dir(unpack_dir, basename: Pathname(basename), verbose: verbose || false)
   end
 
   def extract_nestedly(to: nil, basename: nil, verbose: false, prioritise_extension: false)
diff --git a/Library/Homebrew/unpack_strategy.rbi b/Library/Homebrew/unpack_strategy.rbi
new file mode 100644
index 0000000000..d7550c56b2
--- /dev/null
+++ b/Library/Homebrew/unpack_strategy.rbi
@@ -0,0 +1,5 @@
+# typed: strict
+
+module UnpackStrategy
+  include Kernel
+end
diff --git a/Library/Homebrew/unpack_strategy/zip.rb b/Library/Homebrew/unpack_strategy/zip.rb
index 09db337f84..ac5238d5ca 100644
--- a/Library/Homebrew/unpack_strategy/zip.rb
+++ b/Library/Homebrew/unpack_strategy/zip.rb
@@ -1,27 +1,31 @@
-# typed: true
+# typed: strict
 # frozen_string_literal: true
 
 module UnpackStrategy
   # Strategy for unpacking ZIP archives.
   class Zip
+    extend T::Sig
+
     include UnpackStrategy
 
     using Magic
 
+    sig { returns(T::Array[String]) }
     def self.extensions
       [".zip"]
     end
 
+    sig { params(path: Pathname).returns(T::Boolean) }
     def self.can_extract?(path)
       path.magic_number.match?(/\APK(\003\004|\005\006)/n)
     end
 
     private
 
-    def contains_extended_attributes?(path)
-      path.zipinfo.grep(/(^__MACOSX|\._)/).any?
+    sig do
+      override.params(unpack_dir: Pathname, basename: Pathname, verbose: T::Boolean)
+              .returns(SystemCommand::Result)
     end
-
     def extract_to_dir(unpack_dir, basename:, verbose:)
       quiet_flags = verbose ? [] : ["-qq"]
       result = system_command! "unzip",