diff --git a/CHANGES b/CHANGES
index d19822de1..aca79be5d 100644
--- a/CHANGES
+++ b/CHANGES
@@ -123,6 +123,7 @@ Bugs fixed
 * #7668: autodoc: wrong retann value is passed to a handler of
   autodoc-proccess-signature
 * #7711: autodoc: fails with ValueError when processing numpy objects
+* #7791: autodoc: TypeError is raised on documenting singledispatch function
 * #7551: autosummary: a nested class is indexed as non-nested class
 * #7661: autosummary: autosummary directive emits warnings twices if failed to
   import the target module
diff --git a/sphinx/ext/autodoc/__init__.py b/sphinx/ext/autodoc/__init__.py
index 09db26948..06961d1f8 100644
--- a/sphinx/ext/autodoc/__init__.py
+++ b/sphinx/ext/autodoc/__init__.py
@@ -1222,7 +1222,15 @@ class FunctionDocumenter(DocstringSignatureMixin, ModuleLevelDocumenter):  # typ
 
     def annotate_to_first_argument(self, func: Callable, typ: Type) -> None:
         """Annotate type hint to the first argument of function if needed."""
-        sig = inspect.signature(func)
+        try:
+            sig = inspect.signature(func)
+        except TypeError as exc:
+            logger.warning(__("Failed to get a function signature for %s: %s"),
+                           self.fullname, exc)
+            return
+        except ValueError:
+            return
+
         if len(sig.parameters) == 0:
             return
 
@@ -1811,7 +1819,14 @@ class MethodDocumenter(DocstringSignatureMixin, ClassLevelDocumenter):  # type:
 
     def annotate_to_first_argument(self, func: Callable, typ: Type) -> None:
         """Annotate type hint to the first argument of function if needed."""
-        sig = inspect.signature(func)
+        try:
+            sig = inspect.signature(func)
+        except TypeError as exc:
+            logger.warning(__("Failed to get a method signature for %s: %s"),
+                           self.fullname, exc)
+            return
+        except ValueError:
+            return
         if len(sig.parameters) == 1:
             return