These seem to be the rules:
Commons:
4 of each common in a case with very, very few exceptions.
There is significant correlation between certain commons and the picture on the booster cover. For example, if you buy a Hezrou picture booster, you'll likely get a dwarf raider.
Uncommons:
You'll likely get 1 or 2 of each uncommon in a case, though in rare instances there have been cases with 0 or 3 of a certain uncommon.
There is supposed to be 1 large uncommon in every booster.
Rares:
You are very likely to get 12 different rares in a case, though occasionally there is a repeat.
There is no correlation of rares between cases, so you're unlikely to get a complete set just by buying 2, 3, or 4 cases.
Seeding:
The seeding for uncommons and commons appears to be as follows (meaning that you're supposed to get 1 from each of the following groups, plus a rare, in each booster you open):
Commons:
Group 1: 1, 29, 44 (Caravan Guard, Wild Elf Raider, Kobold Soldier)
Group 2: 31, 42, 51 (Dwarf Mercernary, Hobgoblin Impaler, Abyssal Skulker)
Group 3: 30, 50, 57 (Xeph, Skeletal Archer, Ophidian)
Group 4: 3, 38, 60 (Dwarf Raider, Direguard, Wrackspawn)
Uncommons:
Group 1 (larges): 13, 20, 28, 32, 37, 53, 54, 59 (Werebear, Large Air Elemental, Weretiger, Blackscale Lizardfolk, Chraal, Feral Minotaur, Fiendish Dire Wolverine, Troll Slasher)
Group 2: 4, 12, 18, 24, 40, 43, 49, 56 (Dwarf Wizard, Village Priest, Elven Swashbuckler, Talenta Halfling, Flamebrother Salamander, Imp, Vargouille, Magmin)
Group 3: 2, 5, 7, 22, 25, 27, 34, 52 (Cleric of Dol Arrah, Justice Archon, Spiker Champion, Longstrider Ranger, Thorn, Wand Expert, Scorpion Clan Drow Fighter, Bugbear Champion of Erythnul)