The first thing you need to know is that a crusher's durability comes not from high hp totals, but from steadily increasing deflection chance as they age.
Young Crusher's aren't shit, mature aren't nearly as good as elite, and the biggest change from elite to ancient is melee damage.
I'll be going through by modes of attack.
Specialist weapons:
Smartgun - Currently shit, at the range that smartguns are firing you wont do significant damage.
SADAR - AP rounds will fuck even an ancient crusher, but you need to actually hit it. The best way to do that is to wait until the crusher is retreating, wounded, and then pop the sucker like a grape. HE rounds do nothing, don't bother.
Sniper - These rounds can be deflected (need to test after recent changes), and a few incendiary shots without time to heal will kill mature, probably elite.
Flamethrower - does excellent damage to a crusher, and again, crushers don't have much hp compared to how durable they seem. A few flamer blasts and they are heavily injured or dead.
Standard weapons:
Note: AP ammo (as of 8-8) has been buffed significantly and now forms an excellent counter against crushers. Too bad marines always use it up before mature/elite crushers emerge

Rifles - Pretty much all the same shit. Great damage from behind, ok from the sides, at elite won't be effective from the front at all. For the most part you're just wasting your ammo with this unless you have AP rounds or fighting a young crusher.
Shotguns - Slugs are very effective, on par with AP ammo. Even at long range still has a decent chance to hit, and a handful of shots will kill a young or mature crusher. Especially incendiary slugs.
Buckshot does /not/ stun a crusher at all, but at 1 tile between you and the crusher has a good chance of landing a few pieces of buckshot. This will take out the majority of a crusher's health if you're lucky.
SMG - Are you kidding?
Revolver - better than a smg, which isn't saying much.
Mines - these don't damage a crusher, but they blind it for a few seconds, and it /cannot/ moonwalk during this time. If you want to kill one, bait it with a mine and then lay into it while the crusher is blinded with weapons that pierce armor.
Sentries - High chance of deflection at long range. Lower chance up front, but a crusher charge will knock it down and then it'll get swarmed.
Notable mentions:
Dragonov fires sniper rounds. Kills crushers extremely well.
TL;I'm a baldie
1) Shotgun good, rifle bad. Slugs good, buckshot good but risky.
2) Fire good, grenades bad
3) Mines don't damage but they blind. Take advantage.
4) AP ammo is good now, but don't waste it on young crushers or T1 xenos.