This question extends What are some natural arithmetical statements independent of ZFC? beyond the realm of just number theory.
Scott Aaronson has pointed out that it's surprising how rarely the "Godelian gremlin" rears its head - that is, it's very rare for a problem of mathematical interest to end up being undecidable within ZFC, without having been specifically constructed to be. What problems/conjectures were proposed strictly before being shown to be independent of ZFC? The only examples that I'm aware of are Hilbert's second problem, the continuum hypothesis, Suslin's problem, Kaplansky's conjecture on Banach algebras, and the Whitehead problem. (Five examples may seem like a lot, but this is of course a vanishingly small fraction of all mathematical problems that have ever been posed.)