Destruction:
Flames: Flamethrower spell
Fireball: 75 points of damage in a 15 foot radius
Firestorm: Explosion centered around caster doing 75 damage, closer enemies get hurt more
Ice spike
Frostbite: emits a stream of frost similar to flames, that also drains fatigue.
Ice Storm: (most likely a reskin of firestorm)
Frost Rune: Place a exploding frost mine on the ground
Chain Lightning : 50 damge, and double that damage to magika, also bounces off objects.
Spark
Lightning Bolt
Conjuration:
Bound Battle Axe: summon a glowing magic battle axe
Bound Sword : summon a glowing magic sword
Summon Frost Atronach
Summon "ghost" Wolf
Soul Trap
Restoration:
Healing: Heals as long as the button is pressed
Magical Wards
Circle of Protection: Creates a holy circle on the ground repelling enemies nearby ( only seen being used on undead).
Turn undead
illusion:
Light
Frenzy: moderately powerful creatures and people will attack all nearby for 60 seconds
Clairvoyance: Creates a blue line on the ground leading you to your current objective
Charm
Invisibility
Alteration:
Detect Life: nearby living creatures ( but not undead, machines of Daedra) can be seen through walls
Telekinesis
Water breathing
Paralysis
Shouts:
Slow Time
Unrelenting Force
Fire breath
Ice form
Whirlwind Sprint
Storm call
Animal Allegiance
Note: all spells appear to be separate from each other ( such as flames and fireball )