I think we should also mention which enemies have to be fought in each wave. Maybe there will be contributions on which weapon to use (in case one isn't limited by the ribbon challenge) for each type of enemies. Especially for the Siren (at least i couldn't kill here before she summoned some allies)
1st Wave: Soldiers
2nd Wave: Soldiers (taking the infusion out of the supply machine and using it for upgrading your shield might help)
3rd Wave: Soldiers
Here are further Ribbon Challenges for:
4th Wave (Soldiers and Volley Gunners): Defeat the wave using only machine guns or the crank gun - Not hard at all. The soldiers shouldn't be a problem, the volley gunner(s) can be taken out from afar.
5th wave (Handyman): Defeat the wave using only the pistol and hand cannon - equip both the hand cannon and the pistol - although i preferred the hand cannon, since it does way more damage - before leaving to the battle (you could also upgrade your hand cannon and/or pistol to increse damage). This way you can't accidently attack the handyman with any other gun. now keep shooting the handyman at his heart, but be sure not to use any vigors or the challenge fails.
6th wave (Soldiers and RPG Soldiers): Defeat the wave without picking up weapons or looting corpses - Easy - just don't pick up any weapons or loot the corpses before the wave ends. You can still loot the corpses after the wave is finished and the challenge is complete before returning to the armory. You can also try to get the 'Missile Defense Sysmte' achievement here. Just keep the RPG soldier as the last enemy and try to shoot his missiles from afar right after he launches them at you. The machine gun might be the best choice here.
7th wave (Soldiers, Snipers and Zealots): Defeat all enemies using only zoomable weapons while they are zoomed - I tried the Sniper Rifle (of course) and the machine gun. Both work towards this challenge. Just be shure that you click your right stick to go into aiming mode before shooting!
8th wave (Zealots and Volley Gunners): Defeat all enemies using only Tears; Possession Traps; and Return to Sender traps - I would recommend you to stick to the Traps and the Possession vigor, since you would need to buy the Return to Sender vigor to use it. Bringing in the Rocket Automation or the Tesla Coil suited me perfect, and a few possession traps. Easy method: bring in the Tesla Coil through a tear and keep running around it. The Zealots and Volley Gunners will be shocked by the Tesla Coil permanently until their death ;-)
9th wave (Soldiers and Handyman): Defeat all enemies without ever hitting anyone in the head or the heart - This one's a bit tricky. I don't know if you can shoot the soldiers into their chest with a shotgun and not hitting their heart, or if the heart mention is only considering the handyman (needs confirmation). Anyway: don't shoot the soldiers head's (use skyline attacks) and the handymans head and heart (use the Patriot or the Decoy, so you can get a good shot at him)!
10th wave (Soldiers and Siren): Defeat the Siren before she performs a resurrection - I have no idea how one could kill the Siren before her performing a resurrection
So far, so good. Will keep adding my ideas when I have time again.