Fix docs

Author: ianthomas23

doc/api/next_api_changes/deprecations/27719-IT.rst

@@ -1,8 +1,8 @@
-`~.rcsetup.interactive_bk`, `~.rcsetup.non_interactive_bk` and `~.rcsetup.all_backends`
+``rcsetup.interactive_bk``, ``rcsetup.non_interactive_bk`` and ``rcsetup.all_backends``
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 ... are deprecated and replaced by
-:meth:`matplotlib.backends.registry.backend_registry.list_builtin`
+:meth:`matplotlib.backends.registry.BackendRegistry.list_builtin`
 with the following arguments
 
 - `matplotlib.backends.registry.BackendFilter.INTERACTIVE`

doc/users/next_whats_new/backend_registry.rst

@@ -1,5 +1,6 @@
 BackendRegistry
 ~~~~~~~~~~~~~~~
 
-New `~/.BackendRegistry` class is the single source of truth for available
-backends. The singleton instance is `matplotlib.backends.registry.backend_registry`.
+New :class:`~matplotlib.backends.registry.BackendRegistry` class is the single
+source of truth for available backends. The singleton instance is
+``matplotlib.backends.registry.backend_registry``.

lib/matplotlib/backends/registry.py

@@ -3,7 +3,7 @@
 
 class BackendFilter(Enum):
     """
-    Filter used with :meth:`~.BackendRegistry.list_builtins`
+    Filter used with :meth:`~matplotlib.backends.registry.BackendRegistry.list_builtin`
 
     .. versionadded:: 3.9
     """
@@ -17,6 +17,9 @@ class BackendRegistry:
 
     This is the single source of truth for available backends.
 
+    All use of ``BackendRegistry`` should be via the singleton instance
+    ``backend_registry``.
+
     .. versionadded:: 3.9
     """
     # Built-in backends are those which are included in the Matplotlib repo.