James Harbeck laid out the case against them a few years ago in an article that, to make his point, lacks apostrophes:
Why are so many people so confused by apostrophes? Because they cant hear them in speech, and they dont serve a valuable grammatical function. They simply mark contraction or possession, and you can tell the meaning without them. If you couldnt, the indignant red-pen-wielding self-appointed correction brigades wouldnt know for sure which ones were wrong because the meaning wouldnt be clear. But they always do know, because the meaning is clear even when the apostrophe is used wrongly or omitted.
I liken apostrophes to library fines. Fines are an outmoded practice based on faulty assumptions, and they annoy patrons and staff equally. When libraries do get rid of them, patron satisfaction increases and items miraculously still come back.
Same with apostrophes. They no longer serve any practical function, are too easily misused, and, most importantly, are a pain to type on a typewriter. English would be better off without them. So lets get rid of them.