What abilities allow you to break control impairing effects and other conditions?

Solution 1:

Monk: Serenity

You are enveloped in a protective shield that absorbs all incoming damage for 3 seconds and grants immunity to all control impairing effects (4 seconds with Ascension rune).

Witch Doctor: Spirit Walk

Leave your physical body and enter the spirit realm for 2 seconds. While in the spirit realm, your movement is unhindered.

Your link to the spirit realm will end if your physical body sustains 50% of your maximum Life in damage.

Demon Hunter: Smokescreen

Vanish behind a wall of smoke, becoming momentarily invisible for 1 second (without Lingering Fog rune, 1.5 with).

Wizard: Mirror Image

Summon 2 illusionary duplicates of yourself that last for 7 seconds and have 25% of your Life. The images may cast some of the same spells as you, but those spells deal no damage.

Wizard: Teleport with Fracture

Normally you can teleport when you are Jailed, but you will still be jailed after Teleporting. With the Fracture rune the Teleport will function like using Mirror Images, breaking the jail. Teleport will not however work while Frozen, like Mirror Images will,b ecause for the Images to spawn you need to Teleport first.

Barbarian: Furious Charge

Rush forward knocking back enemies and inflicting 195% weapon damage to enemies along the path of the charge.

Barbarian: Wrath of the Berserker

Enter a berserker rage which raises several attributes for 15 seconds.

And related to using abilites while supressed, the Wizard can activate Diamond Skin even under CC.

Solution 2:

The Witch Doctor has Spirit Walk. I've only used it to break freeze, though.

Solution 3:

Demon Hunter has Smoke Screen, allowing to vanish behind a wall of smoke, becoming momentarily invisible. It works very well for escaping most enemy traps.

Solution 4:

Specifically for the Jailed effect, the Monk's Dashing Strike ability allows you to move a short distance while within the jail.

You're still stuck in the jail, though.