1. (Soldiers; 6 enemies): Defeat all enemies with the sky-hook melee – Not too hard, since it’s only soldiers you have to fight. Equipping Gear which grants Melee damage will help a lot (e.g. Burning Halo); and it doesn’t not fail the challenge, if the enemy dies from the elemental damage done by the Gear!
2. (Soldiers; 7 enemies): Defeat all enemies with the pistol – Upgrading your Pistols damage and Accuracy prior to starting the wave (if not done yet) might help here. Apart from that, pretty self-explanatory.
3. (Soldiers, Volley Gunners; 7 enemies ): Defeat all enemies using only the Undertow Vigor – Again, not very hard. Equip your Undertow Vigor and wait for the soldiers to be near an edge where you can ‘knock’ them off easily.
4. (Soldiers, RPG Soldiers, Zealots; 6 enemies): Defeat all enemies with vigor traps – Remember that you can only set traps with Possession, Shock Jockey, Murder of Crows, Devil’s Kiss and Return to Sender. Take care not to line up 2 enemies in front or behind you. If they accidently shoot each other, you will fail this challenge. SO I would suggest not to use a Possession trap!
5. (Soldiers, Handyman; 5 enemies): Defeat the Handyman before any other enemy – Self-explanatory, just be sure not to use Undertow on the Handyman when other enemies are nearby. The chance to set them flying off is too high.
6. (Soldiers, Patriots;7 enemies): Defeat 5 enemies via friendly fire – Lower the health of the soldiers and then use your Possession Vigor on that Patriot to let him kill 5 of the Soldiers (you need a lot of Salt here to keep that Patriot possessed).
7. (Soldiers, Snipers, Volley Gunners + Zealot(!); 6 enemies): Defeat the Volley Gunner and the Zealot before damaging any snipers – The Zealot wasn’t mentioned in the ‘Get ready to fight’-List, but there is one. Just make sure you kill everyone but the snipers first. There are 2 or 3 Snipers in that wave.
Here's a picture for those of you having trouble to identify the snipers (well, it's a dead one)
8. (Soldiers, RPG-Soldiers; 7 enemies): Defeat all enemies with turrets or environmental hazards – Just as the challenge reads: Use only Rocket or Gun Automations or set Oil spills on Fire or shock Water Puddles. Tesla Coil is also fine (might need confirmation).
9. (Soldiers, RPG-Soldiers, Firemen; 7 enemies): Defeat all enemies in 60 seconds using vigors only – Make sure your salts are always full and then keep throwing the enemies off of Columbia with Undertow.
10. (Soldiers, RPG-Soldiers, Firemen; 8 enemies): Take no damage for 30 seconds – Just as JuSt BLaZe has mentioned. Stay still from where you start and keep watching the enemies as they move. If they seem to be looking at you, just run for some cover and hide. Shouldn’t be too hard.
11. (Soldiers, RPG-Soldiers, Volley Gunners, Firemen; 6 enemies): Defeat all enemies with the Shotgun – Self-explanatory
12. (Soldiers, Firemen; 8 enemies): Defeat the wave without pinking up weapons or looting corpses – As in the previous challenges in Duke and Dimwitt, don’t pick up weapons or loot corpses until the challenge is completed (wave is over). You can get rid of the Firemen easily by throwing them off of Columbia with Undertow.
13. (Snipers, Patriots; 4 enemies): Defeat all enemies while they are distracted with a Booker decoy tear – Simple enough, just set up that decoy ( there are 2 in this wave) and wait for the enemies to get to it, then use your Undertow Vigor and throw them off of Columbia (this works for the 2 soldiers only)! For the Patriots: You can shoot them down to low health, set up the decoy and then destroy them.
14. (Soldiers, RPG-Soldiers; 8 enemies): Defeat each enemy with the same weapon-type it wields – That one’s a bit tricky. For the RPG-soldiers, use an RPG to kill them, for the Soldiers, you have to use a machine gun and a shotgun. The easiest way to do this might be to set up a decoy and check which weapon each enemy is carrying. Then take them out one after another. Remember, you can use tears to get the specific weapon needed to kill the enemies.
15. (Firemen, Handyman;3 enemies): Defeat each enemy with a different weapon – You can kill the Firemen with Undertow by knocking him off of Columbia. The 2 Handyman have then to be killed with different weapons.