diff --git a/Library/Homebrew/livecheck/strategy.rb b/Library/Homebrew/livecheck/strategy.rb
index 0040bc4821..b77610d0c4 100644
--- a/Library/Homebrew/livecheck/strategy.rb
+++ b/Library/Homebrew/livecheck/strategy.rb
@@ -86,6 +86,7 @@ module Homebrew
       # At present, this should only be called after tap strategies have been
       # loaded, otherwise livecheck won't be able to use them.
       # @return [Hash]
+      sig { returns(T::Hash[Symbol, T.untyped]) }
       def strategies
         return @strategies if defined? @strategies
 
@@ -107,8 +108,9 @@ module Homebrew
       # @param symbol [Symbol] the strategy name in snake case as a `Symbol`
       #   (e.g. `:page_match`)
       # @return [Strategy, nil]
+      sig { params(symbol: T.nilable(Symbol)).returns(T.nilable(T.untyped)) }
       def from_symbol(symbol)
-        strategies[symbol]
+        strategies[symbol] if symbol.present?
       end
 
       # Returns an array of strategies that apply to the provided URL.
@@ -119,7 +121,16 @@ module Homebrew
       # @param regex_provided [Boolean] whether a regex is provided in the
       #   `livecheck` block
       # @return [Array]
-      def from_url(url, livecheck_strategy: nil, url_provided: nil, regex_provided: nil, block_provided: nil)
+      sig {
+        params(
+          url:                String,
+          livecheck_strategy: T.nilable(Symbol),
+          url_provided:       T::Boolean,
+          regex_provided:     T::Boolean,
+          block_provided:     T::Boolean,
+        ).returns(T::Array[T.untyped])
+      }
+      def from_url(url, livecheck_strategy: nil, url_provided: false, regex_provided: false, block_provided: false)
         usable_strategies = strategies.values.select do |strategy|
           if strategy == PageMatch
             # Only treat the `PageMatch` strategy as usable if a regex is
@@ -143,6 +154,7 @@ module Homebrew
         end
       end
 
+      sig { params(url: String).returns(T::Array[T::Hash[String, String]]) }
       def self.page_headers(url)
         headers = []
 
diff --git a/Library/Homebrew/livecheck/strategy/apache.rb b/Library/Homebrew/livecheck/strategy/apache.rb
index 36b2395094..6cd4e29416 100644
--- a/Library/Homebrew/livecheck/strategy/apache.rb
+++ b/Library/Homebrew/livecheck/strategy/apache.rb
@@ -37,6 +37,7 @@ module Homebrew
         #
         # @param url [String] the URL to match against
         # @return [Boolean]
+        sig { params(url: String).returns(T::Boolean) }
         def self.match?(url)
           URL_MATCH_REGEX.match?(url)
         end
diff --git a/Library/Homebrew/livecheck/strategy/bitbucket.rb b/Library/Homebrew/livecheck/strategy/bitbucket.rb
index 4072748aed..a3fc074af2 100644
--- a/Library/Homebrew/livecheck/strategy/bitbucket.rb
+++ b/Library/Homebrew/livecheck/strategy/bitbucket.rb
@@ -44,6 +44,7 @@ module Homebrew
         #
         # @param url [String] the URL to match against
         # @return [Boolean]
+        sig { params(url: String).returns(T::Boolean) }
         def self.match?(url)
           URL_MATCH_REGEX.match?(url)
         end
diff --git a/Library/Homebrew/livecheck/strategy/cpan.rb b/Library/Homebrew/livecheck/strategy/cpan.rb
index 88965bd8d6..f1fd982822 100644
--- a/Library/Homebrew/livecheck/strategy/cpan.rb
+++ b/Library/Homebrew/livecheck/strategy/cpan.rb
@@ -35,6 +35,7 @@ module Homebrew
         #
         # @param url [String] the URL to match against
         # @return [Boolean]
+        sig { params(url: String).returns(T::Boolean) }
         def self.match?(url)
           URL_MATCH_REGEX.match?(url)
         end
diff --git a/Library/Homebrew/livecheck/strategy/git.rb b/Library/Homebrew/livecheck/strategy/git.rb
index 55c3c05562..d68853ab9b 100644
--- a/Library/Homebrew/livecheck/strategy/git.rb
+++ b/Library/Homebrew/livecheck/strategy/git.rb
@@ -50,6 +50,7 @@ module Homebrew
         # @param url [String] the URL of the Git repository to check
         # @param regex [Regexp] the regex to use for filtering tags
         # @return [Hash]
+        sig { params(url: String, regex: T.nilable(Regexp)).returns(T::Hash[Symbol, T.untyped]) }
         def self.tag_info(url, regex = nil)
           # Open3#capture3 is used here because we need to capture stderr
           # output and handle it in an appropriate manner. Alternatives like
diff --git a/Library/Homebrew/livecheck/strategy/github_latest.rb b/Library/Homebrew/livecheck/strategy/github_latest.rb
index 68a9657448..c575bf55bb 100644
--- a/Library/Homebrew/livecheck/strategy/github_latest.rb
+++ b/Library/Homebrew/livecheck/strategy/github_latest.rb
@@ -52,6 +52,7 @@ module Homebrew
         #
         # @param url [String] the URL to match against
         # @return [Boolean]
+        sig { params(url: String).returns(T::Boolean) }
         def self.match?(url)
           URL_MATCH_REGEX.match?(url)
         end
diff --git a/Library/Homebrew/livecheck/strategy/gnome.rb b/Library/Homebrew/livecheck/strategy/gnome.rb
index 7af002b9f5..b8bd9621cb 100644
--- a/Library/Homebrew/livecheck/strategy/gnome.rb
+++ b/Library/Homebrew/livecheck/strategy/gnome.rb
@@ -40,6 +40,7 @@ module Homebrew
         #
         # @param url [String] the URL to match against
         # @return [Boolean]
+        sig { params(url: String).returns(T::Boolean) }
         def self.match?(url)
           URL_MATCH_REGEX.match?(url)
         end
diff --git a/Library/Homebrew/livecheck/strategy/gnu.rb b/Library/Homebrew/livecheck/strategy/gnu.rb
index dac3db3ef6..71c778bc0a 100644
--- a/Library/Homebrew/livecheck/strategy/gnu.rb
+++ b/Library/Homebrew/livecheck/strategy/gnu.rb
@@ -44,6 +44,7 @@ module Homebrew
         #
         # @param url [String] the URL to match against
         # @return [Boolean]
+        sig { params(url: String).returns(T::Boolean) }
         def self.match?(url)
           URL_MATCH_REGEX.match?(url) && url.exclude?("savannah.")
         end
diff --git a/Library/Homebrew/livecheck/strategy/hackage.rb b/Library/Homebrew/livecheck/strategy/hackage.rb
index 025ebe266a..fe99e66fe5 100644
--- a/Library/Homebrew/livecheck/strategy/hackage.rb
+++ b/Library/Homebrew/livecheck/strategy/hackage.rb
@@ -37,6 +37,7 @@ module Homebrew
         #
         # @param url [String] the URL to match against
         # @return [Boolean]
+        sig { params(url: String).returns(T::Boolean) }
         def self.match?(url)
           URL_MATCH_REGEX.match?(url)
         end
diff --git a/Library/Homebrew/livecheck/strategy/launchpad.rb b/Library/Homebrew/livecheck/strategy/launchpad.rb
index d4f935b90e..ee91769c8c 100644
--- a/Library/Homebrew/livecheck/strategy/launchpad.rb
+++ b/Library/Homebrew/livecheck/strategy/launchpad.rb
@@ -35,6 +35,7 @@ module Homebrew
         #
         # @param url [String] the URL to match against
         # @return [Boolean]
+        sig { params(url: String).returns(T::Boolean) }
         def self.match?(url)
           URL_MATCH_REGEX.match?(url)
         end
diff --git a/Library/Homebrew/livecheck/strategy/npm.rb b/Library/Homebrew/livecheck/strategy/npm.rb
index db3b062247..4faab03ca2 100644
--- a/Library/Homebrew/livecheck/strategy/npm.rb
+++ b/Library/Homebrew/livecheck/strategy/npm.rb
@@ -31,6 +31,7 @@ module Homebrew
         #
         # @param url [String] the URL to match against
         # @return [Boolean]
+        sig { params(url: String).returns(T::Boolean) }
         def self.match?(url)
           URL_MATCH_REGEX.match?(url)
         end
diff --git a/Library/Homebrew/livecheck/strategy/pypi.rb b/Library/Homebrew/livecheck/strategy/pypi.rb
index ecbd14c505..96a5b1fbf7 100644
--- a/Library/Homebrew/livecheck/strategy/pypi.rb
+++ b/Library/Homebrew/livecheck/strategy/pypi.rb
@@ -41,6 +41,7 @@ module Homebrew
         #
         # @param url [String] the URL to match against
         # @return [Boolean]
+        sig { params(url: String).returns(T::Boolean) }
         def self.match?(url)
           URL_MATCH_REGEX.match?(url)
         end
diff --git a/Library/Homebrew/livecheck/strategy/sourceforge.rb b/Library/Homebrew/livecheck/strategy/sourceforge.rb
index a6a9a9add2..0ffcdb0d2e 100644
--- a/Library/Homebrew/livecheck/strategy/sourceforge.rb
+++ b/Library/Homebrew/livecheck/strategy/sourceforge.rb
@@ -47,6 +47,7 @@ module Homebrew
         #
         # @param url [String] the URL to match against
         # @return [Boolean]
+        sig { params(url: String).returns(T::Boolean) }
         def self.match?(url)
           URL_MATCH_REGEX.match?(url)
         end
diff --git a/Library/Homebrew/livecheck/strategy/xorg.rb b/Library/Homebrew/livecheck/strategy/xorg.rb
index 6f6b94b0a4..5fcbbdd449 100644
--- a/Library/Homebrew/livecheck/strategy/xorg.rb
+++ b/Library/Homebrew/livecheck/strategy/xorg.rb
@@ -64,6 +64,7 @@ module Homebrew
         #
         # @param url [String] the URL to match against
         # @return [Boolean]
+        sig { params(url: String).returns(T::Boolean) }
         def self.match?(url)
           URL_MATCH_REGEX.match?(url)
         end