Extreme Minorities under FPTP
I commented a little while ago on the often raised danger of extreme minorities winning under lottery-voting. I feel I should point out that this danger is not restricted to lottery-voting. The people over at Make My Vote Count point to this example, where a BNP candidate is elected under FPTP.
As they point out, two-thirds of the vote were cast against him, but that in itself isn't too surprising - given the number of parties running, a plurality is often attained with just 30-40% of the vote, and Labour won a Commons majority with around 35% of the national vote. If we speculate, however, that the BNP candidate would probably have been last choice, at least of those on the left (if not perhaps UKIP voters), then he could have been defeated by any system that took second preferences into account.
Another possibility would, of course, be to copy the French presidential 'plurality run-off' model. In this case, the BNP candidate would make the second round - much like Le Pen did in 2002 - but similarly would probably lose heavily.