In particular, the question refers to problems dealing with the infinity, hence the notion of "happening", which evokes the reality of an event, is unclear. The answer may depend on what is considered "possible": is it possible to play heads and tails and get heads indefinitely? This is "infinitely unlikely" but conceivable.
These unempty events with zero probability are an abstraction, but often make a more concrete sense as a limit: the above example has zero probability because the probability of getting heads in a row converges to 0 as grows to infinity. This way, only finite spaces are involved. Of course, this is just a way to hide the infinity under the carpet.
Anyway, I may be wrong but I think that the interpretations of probability (like the one you are evoking) mostly refer to finite probability spaces.