My pet peeves when it comes to undecidability:

1. People saying "sure, problem A may be undecidable but it's decidable for instance X". That's not how the concept works - if you restrict the a problem to a specific instance (in the sense of having a decider for the intersection of A and {X}), then it will always be decidable because either the Turing Machine that always outputs true or the Turing Machine that always outputs false is a decider (it doesn't matter that we don't know which one). I know what people are trying to say - "just because a problem is undecidable it doesn't mean you can't solve specific instances" - but this is arguing against a misconception of what decidability means.

2. "Decidability doesn't matter because real-world computers have finite memory". Yeah, but if your solution to problem X involves running an algorithm until you've run through all possible 2^N configurations for N=the size of your memory, you haven't really practically solved it either in any way that anyone would want to use.