Rand al'Thor

 I'll link things and give little synopsises here.

The One Power split into Saidin and Saidar, and Rand specifically would use Saidin, because he's male. He does later become capable of using the True Power, but it has side effects so not going to be a problem.

Weaves or Webs are the names of "spells", as known in the Third Era (Rand's Time) or The Age of Legends (Lews' Time) respectively. They are described in creation to be similar to textiles - you "weave" the magic in a specific pattern and it causes an effect. You can "knot" the weave, and instead of ending, it becomes a continuation effect. Most of the time people just stop focusing on their weave when the intended effect is done, and it dissipates on its own. 

Specific weaves Rand or Lews definitely know are:

• Call Lightning Summon lightning to strike, does not require weather. Does come with all the heat and light lightning strikes naturally do.
• Refresh Healing-like weave that "hides" fatigue. It WILL catch up eventually
• Make Heartstone  Turn an object into a nigh indestructable form of that object, with its original weight and density. Hitting it with One Power only strengthens it. Can resist Balefire.
• Traveling Point to Point teleportation via a "Gate". Requires knowing your starting location very well.
• Skimming Does not require knowing more than your general direction, one person at a time.
• Deathgates When your travelling gate spells have sharp edges, so you rotate it and send it going through an army! surprise sliced chunks of horrible things dropped random places.
• Mirror of Mists Illusion magic, effects size, appearance, voice. Includes words
• Folded Light Illusion magic, effects visibility. Despite claims, Rand probably can't move around and keep it going.
• Disguise Specifically of his ability to use magic.
• Inversion Less of a new spell, and more of turning your spells invisible. Used for Wards mostly, or some kind of trap, or just making it so no one can see what you are casting.
• Arrows of Fire 10 filaments of fire per fingertip that heat whatever they come in contact with to past boiling! They pierce! Up to 100 per cast! Lews that was hella overkill.
• Balefire Erase someone from existence! if you use enough power, you can retroactively un-dead their victims! Side effects include breaking reality if too much is used!
• Bonding Create an enduring psychic link between Rand and someone else. Bond-mate gains heightened abilities (can go without food, water, rest for longer) and damage resistance (no quick shock deaths for them). Gain some ability to sense Shadowspawn. Emotional and physical state sensing between the two, and unerring ability to locate the other's relative direction, and vague distance. HOWEVER, if Rand dies, his bond-mate will go nuts with fatalistic vengeance.
• Cutting Weaves Cuts off attacks from others. Basically interrupting spells.
• Eavesdropping Hear things from a distance, or an adjacent room. Seems to require some airflow.
• Fireball Smaller than the DnD flavor, although it does detonate on impact. Many people mime the fling, it's not necessary. 
• Blossoms of Fire Explosive fling spell. ~100 pace Prime Effective Zone, 30ft explosion diameter. Makes a high pitched whining noise as they move.
• Hardened Air Multipurpose manipulation of Air. It 'stiffens' the air into a 3D object, which can then be used to hit, stab, cut, slice, peel, or as more of a 'tentacle' for moving, grabbing, flinging, or binding people. Can also be used to make a shelter (like Shield of Air)
• Shield of Air Specific kind of Hardened Air weave, can be repel or slow movement, in a simple wall or a dome.
• Protective Cocoon Shield of Air, except air doesn't pass through it, so only can be used for a short time. Repels magic except for Balefire.
• Heating - Air, can keep you warm up on a snowy mountain summit. Liquid, from a hot tea when you need it to refreshing your bath.
• Keeping Preservation spell! Works on food, plants. Will kill a living being if placed on them. Does need a refreshing, depending on how many things it was placed on at that time, and how long ago.
• Liquid Manipulation Attract, repel, collect, generally move liquids. Doesn't include bodily fluids.
• Make Bridge Magical portable bridge. Can be made between two boats.
• Make Ice Magically produced Ice Cubes. Doesn't seem to be made into a size for much use other than maybe dropping 
• Make Light Floating balls of light, can be any color
• Night's Shade Shorten your lifespan to a a few weeks in exchange for greater strength, speed, and the ability to blend in the shadows! He knows about it, he's never used it.
• Riven Earth Earth explosion, tends to be a line, has some fire involved. Does not have to start with him, can run perpendicular.
• Sound Amplification Loud speaker spell.
• Welding Stone Joins stones together. More resistant than traditional brick construction, or concrete.
• Shielding Sealing the magic of others, non-permanently.
• Ward Little spells for shields/repelling external forces/'trap spells'/deter pests/alarms/dream realm protections

It's less of a spell, but there's is also the 'Side Effect' of Drawing Too Much on the One Power. This tends to have disastrous consequences for the surroundings. Manetheren's capital was wiped off the map when the Queen drew too much Power. Lews caused an active volcano to form, and it has remained semi-active (like the Cascades) for 3000 years!