The Rambam says that the Shchita of a Mumar for one aveira is Kosher, just that there needs to be a Shochet who will check his knife before he shechts (as one can assume that he won't work too hard to make the knife clean). The only debate in the Gemara is if a Mumar Lechol hatorah kula's shechita is kosher.
So why are many people so careful about the Yiras Shamayim of the Shochet? I know people who will only eat Chassidisher Shchita, some will only eat meat that was shechted by Shochtim from their Chassidus, some will not eat meat outside their house (while they will eat other products), etc. These are people that will eat national hashgachos for most items.
These people can't consider non-Chassidim to be like a mumar lechol Hatorah as if they were, they couldn't rely on their Hashgachos at all (and wouldn't be able to eat anything with an O-U, etc.)
I am looking for sources (such as, Shulchan Aruch demands ..., Simla Chadasha demands etc.)