The thing about all of this is that with some things (the powers themselves) we willingly suspend disbelief, for others (things that are more familiar in everyday life) it's a lot harder.
For a non L&C example, in one episode of Terminator: The Sarah Connor Chronicles they made three important errors about how blood groups work in about twenty seconds, and the only explanation for one would be that Sarah Connor was NOT John Connor's mother. I immediately picked up on that because I'm a biologist and regular blood donor, whereas the sheer impossibility of time travel doesn't worry me at all.
With L&C I think that the secret identity thing is the big stumbling block for a lot of people because most people are good with faces; I don't have a problem with this because I'm really bad at recognizing people - having said that, if I saw someone every day in two different identities I think the penny would drop sooner or later. The powers, even things like the selective hearing, are more of a 'given' and you have to accept them or the character doesn't work at all.
With the Superman Returns / Supergirl / NCIS crossover I've just written the secret identity thing was a big problem - they're detectives, surely they'll notice sooner or later. I was going to get really elaborate about disguises, but in the end I said the hell with it - they get to see Supergirl in civilian clothes and aren't fooled, because she isn't trying to fool them, but right at the end of the story one of them literally bumps into her, in her secret identity, and is completely clueless. It's a Kryptonian thing, maybe they have the same Someone Else's Problem field that stops anyone noticing the TARDIS.
