I am moving & after taking down many pictures, mirrors, etc., I have many plastic anchors left in the sheetrock. What is the best way to patch them?
Depending on the design of them, if they're the type that can be removed ( pulled straight out with needle nose pliers) I would yank them then patch them.
If they are the type that don't want to come out easily, sometimes you can take a small ball peen hammer ( using the rounded end )and lightly tap them below surface level... This way, patching can be accomplished.
And finally, if you can't do anything with them ( in rare cases ) you can drill them out gently, then of course patch.