Credit to Lambstein (Xbox One TA guide) and vSully (Xbox 360 TA guide) for their guides that gave me a headstart on this one.
First: Not all challenges are available immediately and are first listed as (undiscovered), only becoming available after meeting certain requirements in other related challenges; however any progress you did for it while it was hidden DOES get recorded properly.
Second: NOT ALL CHALLENGES APPLY. Any Location-based Challenges and any DLC-related Challenges DO NOT APPLY. All you have to do are the non-location, base-game Challenges.
Most of the challenges are self-explanatory. Here is the list of the level 1 requirement for each (which is all you need, other than the two that must hit level 5 to unlock related Challenges) . Any that are unlocked have an asterisk (*), and if I know the unlock requirement that detail is included under the challenge. Most don't have the actual description, but that means I got them over time before I started keeping track anyways, so it's a good chance you won't have to work at it either. It seems all but two challenges are unlocked by clearing just the first level of another related challenge.
Also, sorry they aren't in a particular order... it seems like they change order sometimes? I dunno.
Low Gravity
Boosted: Perform 50 air boosts
Slampage!: Kill 5 enemies with slam attacks
Death from Above: Kill 50 enemies wile in the air
Eagle Eye: Get 50 critical hits while in the air
Dragon Punch: Deal 5000 melee damage while in the air
Roger Slamjet: Deal 2000 damage with slam attacks
Grinder
Greater Than the Sum of its Parts: Obtain 20 Luneshine weapons from the Grinder (Only available via Moonstone Grind)
This Time for Sure: Perform 10 Moonstone grinds
The Daily Grind: Perform 50 standard grinds
Master Chef: Discover 2 grinder recipes
-------Grinder Notes-------
-Moonstone Grinds have different costs, based on weapon rarity+Unique+recipe (for White, Green, and Blue: either 2+ of the same kind or 3 different).
--White: 49
--Green: 49/75
--Blue: 49/56/62/112
-- 2 Purple+1 Legendary: ?? (I could get the option to appear, but not usable for some reason)
--(I've also seen a 169 cost, but forgot to record what the recipe was)
-The recipe challenge is easy, they are just any different combination of stuff you toss in (if curious, check the wiki:
https://borderlands.fandom.com/wiki/Grinder).
-Luneshine seems to have about a 50% chance of happening, so it may take around 40 Moonstone Grinds to accomplish. That is a minimum moonstones of 980 (100% Luneshine), 1960 (50% Luneshine), or more if you have worse luck. And that's if you limit your Moonstone Grinds to costs of 49 each time.
-------End Grinder Notes-------
Enemies
Dose of Death: Kill 10 enemies infected with space hurps
Not So Tough Now: Kill 25 Eridian Guardians
Slayer of Titans: Kill 1 kraggon badass
Aim for the Little Ones: Kill 10 Opha's spawned Putti
More Where That Came From: Kill 2 Eridian Guardian bosses
No Suit For You!: Kill 10 Lost Legion powersuits
Get Some!: Kill 50 Lost Legion Infantry
Crash & Burn: Destroy 1 Lost Legion jet fighter
Kiss My Badass: Kill 5 Lost Legion badasses
Gift That Keeps Giving: Kill 1 Swagman
Exterminator: Kill 25 rathyds
*The One That Says B.A.M.F: Kill 10 Scav Badasses
---Unlocked after ???
It Wasn't Yours, Anyway: Kill 1 scav jet fighter
Die, Moon Jerks: Kill 75 scavs
Space Dead: Kill 25 scav outlaws
That Was Unpleasant: Kill 25 shugguraths
Torkin' Out the Trash: Kill 5 tork badasses
Big Game Hunt: Kill 25 kraggons
Pest Control: Kill 25 torks
Elemental
Way of the Enkindling: Light 25 enemies on fire
Out with a Bang: Kill 20 enemies with explosive damage
Frost Bite: Deal 5000 damage with cryo DoT effects
Watt's Up?: Deal 5000 damage with shock DoT effects
Chemical Burn: Deal 2500 damage with corrosive DoT effects
Some Like it Hot: Deal 5000 damage with incendiary DoT effects
Toxic Takedown: Kill 20 enemies with corrosive damage
Loot
Junkyard Dog: Loot or purchase 50 white items
A Chunk of Purest Green: Loot or purchase 25 green items
Rare as Rocking Horse...: Loot or purchase 5 blue items
Purple Haze: Loot or purchase 3 purple items
Aaaaaand OPEN!: Open 10 treasure chests
Scrounging Around: Open 50 lootable crates, lockers, and other objects
The Happiest Color: Loot or purchase 1 orange item
One for Every Occasion: Pick up or purchase 10 weapons
Money and Trading
Impulse Shopper: Buy 1 Item of the Day
Mom Would Be Proud: Save 10000 dollars
Over the Moon: Purchase 25 items with Moonstones
Pawn Broker: Sell 50 items to vending machines
Mr. Money Pits: Collect 5000 dollars from cash drops
Trade Negotiations: Trade 1 time with another player
Vehicle
Fender Bender: Kill 5 enemies by ramming them with a vehicle
Road Warrior: Kill 5 enemies while in a vehicle
Improvise, Adapt, Overcome: Kill 10 enemies with a turret or vehicle-mounted weapon
Splat 'n' Slide: Kill 1 enemy by power-sliding over it in a vehicle
Pancakes for Breakfast: Kill 1 enemy with a Stingray slam
Health and Recovery
*Rollin' the Ice: Get 1 second wind by shattering a frozen enemy
---Unlocked after ???
Better You Than Me: Get 10 second winds by killing an ememy
Doctor Feels Good: Recover 1000 health
There Can Be Only... Me: Get 1 second wind by killing a badass enemy
*The Phoenix: Get 1 second wind by killing an enemy with a burn DoT
---Unlocked after ???
Up Unt at Zem: Revive a co-op partner 5 times
*Had a Bit of a Shocker: Get 1 second wind by killing an enemy with a shock DoT
---Unlocked after ???
*Soup's Up: Get 1 second wind by killing an enemy with a corrosive DoT
---Unlocked after ???
Grenades
Betty Boom: Kill 10 enemies with Bouncing Betty grenades
Pass the Chianti: Kill 10 enemies with Transfusion grenades
Sprayowee: Kill 25 enemies with Area-of-Effect grenades
Home Nade Cookin': Kill 10 enemies with grenades
See Ya on the Other Side: Kill 10 enemies with singularity grenades
Big MIRV: Kill 10 enemies with MIRV grenades
Shields
Nova Say Die: Kill 5 enemies with a Nova shield
Ammo Eater: Absorb enemy ammo 20 times with an Absorption shield
Amplitude Killuation: Kill 5 enemies while buffed by an Amplify shield
That'll Learn Ya: Kill 5 enemies with reflected damage from a Spike Shield
Wet Work: Kill 5 enemies while buffed by a Maylay shield
Rocket Launcher
*Missile Magnet: Kill 5 enemies with direct hits from rocket launchers
---Unlocked after completing level 5 in Collateral Damage (200 kills with rocket launcher splash damage)
*Collateral Damage: Kill 5 enemies with rocket launcher splash damage
---Unlocked after ???
Magic Missile: Get 2 second winds with rocket launchers
Get a Rocket Up Ya: Kill 10 enemies with rocket launchers
*Punker Buster: Kill 5 shielded enemies with one rocket each
---Unlocked after ???
*Hand of God: Kill 25 enemies from long range with rocket launchers
---Unlocked after ???
Sniper Rifle
*Ol' Skool: Kill 5 enemies with sniper rifles without using the scope/ironsights
---Unlocked after ???
Clean and Simple: Kill 5 unaware enemies with sniper rifles
*Penetrating Wound: Kill 5 shielded enemies with one shot using sniper rifles
---Unlocked after ???
*Melon Splitter: Get 25 critical hits with sniper rifles
---Unlocked after ???
Sharp Shooter: Kill 20 enemies with sniper rifles
Windage Adjustment: Get 2 second winds with sniper rifles
*Critical Reception: Kill 10 enemies with critical hits using sniper rifles
---Unlocked after ???
Assault Rifle
*Crouch Potato: Kill 25 enemies with assault rifles while crouched
---Unlocked after ???
Assaulty Dog: Get 5 second winds with assault rifles
Assault with a Deadly Weapon: Kill 25 enemies with assault rifles
Aim to Please: Get 25 critical hits with assault rifles
*Hot Lead Injection: Kill 10 enemies with critical hits using assault rifles
---Unlocked after ???
Laser
Aggressive Lasik: Get 25 critical hits with lasers
*Light 'em Up: Kill 10 enemies with critical hits using laser weapons
---Unlocked after ???
Battle Star: Kill 10 flying enemies with laser weapons while in the air
Pew Pew: Kill 25 enemies with laser weapons
I See the Light: Get 2 second winds with laser weapons
SMG
*Dinky Death Dealer: Kill 10 enemies with critical hits using SMGs
---Unlocked after clearing Bring the Pain Lv. 1
Bring the Pain: Get 25 critical hits with SMGs
Nice Spray Job: Kill 25 enemies with SMGs
And Stay Down!: Get 2 second winds with SMGs
Shotgun
*Take It All: Kill 10 enemies from point-blank range with shotguns
---Unlocked after ???
Hello Uncle Buckshot: Get 50 critical hits with shotguns
*Over Achiever: Kill 10 enemies from long range with shotguns
---Unlocked after completing level 5 in Take It All (kill 750 enemies at point-blank with a shotgun)
*Shotty Workmanship: Kill 10 enemies with critical hits using shotguns
---Unlocked after ???
Boomstick Boogie: Kill 25 enemies with shotguns
Shotgunning the Breeze: Get 2 second winds with shotguns
Pistol
Trigger Happy: Kill 25 enemies with pistols
*Gunslinger: Kill 10 enemies shortly after entering ironsights with a pistol
---Unlocked after ???
*Magnum Maestro: Kill 10 enemies with critical hits using pistols
---Unlocked after ???
Pistol Whipped: Get 2 second winds with pistols
Pistoleer: Get 25 critical hits with pistols
Melee
*Captain Cutty: Kill 20 enemies with melee attacks using bladed guns
---Unlocked after ???
Martial Marshal: Kill 25 enemies with melee attacks
General Combat
Action Hero: Kill 25 enemies using your Action Skill
Projectile Proliferation: Fire 5000 rounds
Ice to Meet You: Shatter 20 frozen enemies with weapons
Comet Crash: Shatter 5 frozen enemies with falling damage
Executioner: Kill 20 enemies with critical hits
Having Trouble Breathing?: Shatter 25 enemy oxygen masks
Last Gasp: Kill 10 enemies by asphyxiation
Dark Sider: Kill 25 enemies at night
Day of the Dead: Kill 250 enemies during the day
Barrel of Laughs: Kill 10 enemies with stationary barrels
One Man's Trash: Deal 5000 damage with Tediore reloads
Throw Me the Money: Kill 5 enemies with Tediore reloads
Miscellaneous
The Duelist: Win 1 duel
OC/DC: Complete 5 optional mission objectives
We Have a Contender: Complete 5 challenges
Little on the Side: Complete 10 side missions
Notes:
For any unlockable Challenges, any progress you make for it prior to it unlocking IS tracked, and will be correctly shown when it unlocks.