Ping’s link above has some good material on problems with STV.

Now the reality is that all the systems fail (if they need to provide just one winner) and most of the failures are in somewhat pathological cases, but some fail more than others, so it is better to use systems that fail less often in real cases, as well as systems that are easier to understand. Approval is easiest to understand and implement. Condorcet is also fairly easy to understand (except in ties.) Weighted systems are probably too complex for too many voters.

It is however worth noting that some of the “failures” are definitional. For example, the Condorcet Criterion (that the winner beats everybody else in a head to head race) is not met by many of the non-Condorcet systems. Some systems will choose a winner who is everybody’s 2nd choice and others won’t, and it’s subjective if this is a bug or a feature. (IRV does not choose a candidate who is everybody’s 2nd choice but few people’s first choice, while it does choose that candidate if they are not last in the first round.) Some criteria, like monotonicity make sense to everybody, and IRV fails this. Supporters of IRV just feel it has other virtues to make up for the non-monotonicity. However, IRV is inferior to Condorcet in all criteria. See Voting system.