I think my absolute favorite was Harry Kim never getting promoted (even though everyone around him was) but somehow retaining a critical position on the bridge and the command staff. If he's not worthy of promotion, then bring in someone who is.
I would have voted for Janeway supposedly fiddling with her combadge, but I couldn't have said if that was true or not.
Runners-up: these would actually rate higher than the Harry thing except they're not exactly contradictions, just oversights.
First, B'Elanna reminding Tom that he was expelled from Starfleet Academy even though he wasn't. Look, writers, if you're going to keep confusing him with Nick Locarno then don't pretend he's a different guy. Just pay the royalties and move on. It'll be easier on everyone.
Second, making friends with Captain Korok and his transwarp-capable Borg Sphere and never thinking to ask if he could take them home.
Third, Harry blathering about how witty and charming Irina is even though we never see her change expression. The girl must go through a total metamorphosis when she's off camera.