Barbarian, Bard, Cleric, Druid, Fighter, Monk, Paladin, Ranger, Rogue (Thief), Sorcerer (untrained Wizard), and Wizard.

Single class rules are pretty vanilla, Dualclass is taken out of 3rd Edition in favor of the new Multiclass rules.

Multiclass rules are kind of confusing when read from the PHB, bottom line, you can pretty much multiclass without restriction, you can apply a new level to the class of your choice, though there is a penalty for advancing one class to far in front of the other.

The games mechanics are like this; once you gain a new level you can chose to apply it to your current class or take a new class at 1st level. If you take a new class you gain all the abilities and benefits of a 1st level character in that class, with the restrictions listed on page 56 PHB. You can also find the other restrictions, level benefits, calculation HP and saving throws, and xp penalties listed on page 55-56.

From there you have additional classes called Prestige classes, found in the DM's guide and supplements. Its an interesting idea allowing for a type of multiclassing without an xp penalty. This is more the domain of the DM setup for world or campaign specific classes.

Anyway, the PHB gives a fair run through of the 11 classes and has the new universal xp level chart on page 22. Now everyone uses just one xp table. It also covers probably the only restricting factor when choosing classes, alignment restrictions.

Fighters and Rangers were untouched by the change, but more Barbarians started appearing. The Knight is now treated as multi-alignment Paladins, losing some of their abilities while gaining new ones, these new Paladins were restricted to have the same alignment as their Gods, (otherwise the same as the Paladin in the PHB).

Wizards were also untouched maintaining their specialty schools of Illusion, Necromancy, and Divination. They did find that they could now cast their spells with metal armor on, but with great difficultly. The Elementalists and Shamans found their powers on the rise and are now referred to as Sorcerers.

Clerics, Druids, and Monks had no major changes outside of the spell casting restrictions brought on by the Fall of the Gods, (treat as rules from PHB, see casting limits under the Powers of the Realm section). The Specialty Priest faded in to obscurity as most were lost being sent to search the planes for the shards of the Staff of Godly Might.

Thieves and Bards did not find much changed, but the Thief-Acrobat is now a lost art that no one pursues anymore. The Assassin has since become a Prestige class, (page 29 DMG).

Also available are the Crypt Lord, (page 10-11, Relics and Rituals SB), the Sea Witch,(page18-19, Relics and Rituals SB), and the Vigilant, (page 22-23, Relics and Rituals SB).