One Star Quests
Banned Goods
  • Unlocked: First day
  • Quest Item: Goblot Hairball x 8
  • Reward: 240 gil, Lebreau's Rainbow Tattoo, Tribal Tattoo

Goblot Hairballs are dropped by Goblots. They appear frequently in Dead Dunes, pretty much everywhere you go.
Biologically Speaking
  • Unlocked: First day
  • Quest Item: Desert Scale x 3
  • Reward: 200 gil, Long Gui's Shell

Desert Scales are dropped by Desert Sahagins. You can find them all over the Dead Dunes.
Climbing the Ranks I
  • Unlocked: Third day
  • Quest Item: Dead Man's Teeth x 5
  • Reward: 200 gil, Replica PSICOM Epaulet

Dead Man's Teeth are dropped by Skeletons, found in the Temple Ruins.
Lucky Charm
  • Unlocked: First day
  • Quest Item: Lizard Tail x 3
  • Reward: 400 gil, Pumpkin Head

Lizard Tails are dropped by, well, lizards. They're found periodically as you wander the Dead Dunes; strike them when you find them and they'll drop either ?Unappraised Items or the Lizard Tails you need.
Pride and Greed I
  • Unlocked: Third day
  • Quest Item: Yellowed Skull x 3
  • Reward: 300 gil, Commissioned Pilot's Badge, PSICOM Medic Medal

Yellowed Skulls are found by destroying the skeletal sentries found in the Temple Ruins. Sometimes you'll get ?Unappraised Items instead, so this might take a bit of luck and some retracing of your steps to accomplish.
Pride and Greed II
  • Unlocked: Seventh day
  • Quest Item: Yellowed Skull x 5
  • Reward: 500 gil, Gadot's Blue Emblem, Dark Knight's Shades

Keep on bashin' those statue things.
Supply and Demand
  • Unlocked: Complete Lucky Charm
  • Quest Item: Lizard Tail x 5
  • Reward: 800 gil, Dandy's Mustache, Servant's Mustache

Keep slashing those poor lizards. You can find lots of them by the Oasis Lighthouse.
Two Star Quests
A New Application
  • Unlocked: Complete Family Food
  • Quest Item: Lizard Tail x 20
  • Reward: 1600 gil, Craftsman's Beard, Regent's Mustache

Poor lizards. You'll drive the little creatures to extinction.
A Treasure for a God
  • Unlocked: Complete the Dead Dunes main quest
  • Quest Item: Pilgrim's Crux x 3
  • Reward: 600 gil, Sky Blue Silk Hat

Pilgrim's Cruxes are found all over Dead Dunes. If you can't find them anywhere else, you can buy them from the Crux Hunter in Ruffian.
Climbing the Ranks II
  • Unlocked: Complete Climbing the Ranks I
  • Quest Item: Dead Man's Teeth x 20
  • Reward: 600 gil, PSICOM Mobile Medal

If you beat the last quest, you can do this one. The Temple Ruins have tons of these stupid Skeletons.
Flower in the Sands
  • Unlocked: First day
  • Quest Item: Desert Rose
  • Reward: 700 gil, PSICOM Air Medal

Desert Roses are dropped by Earth Eaters. You can find them near Ruffian; check the Atomos's Sands region in particular to find them with fair regularity.
Heightened Security
  • Unlocked: Fourth day
  • Quest Item: Organic Carapace
  • Reward: 700 gil, Replica PSICOM Emblem, Bandage

Organic Carapaces are dropped by Gurangatches, which can be found in the Lower City of Yusnaan. Hunt all the way to the west, near the edge of the territory, to find them in relative abundance.
Miracle Vintage
  • Unlocked: Fourth day
  • Quest Item: Pot Shard x 10
  • Reward: 500 gil, Yellow Bow Tie, Canned PuPu Logo

You can receive Pot Shards from Schrodingers. They hang out on Central Avenue in Yusnaan, as well as in the Patron's Palace.
Pride and Green III
  • Unlocked: Ninth day
  • Quest Item: Yellowed Skull x 20
  • Reward: 1500 gil, Staff of Judgment, White Guitar

Yeeeep. Bash, bash, bash.
The Real Client
  • Unlocked: Complete Banned Goods and Miracle Vintage
  • Quest Item: Goblot Hairball x 10, Pot Shard x 5
  • Reward: 600 gil, Blue Newsboy Cap, NORA Logo

Goblot Hairballs are dropped by Goblots, which appear regularly in every area of Dead Dunes. Pot Shards are dropped by Schrodingers, which appear occasionally in the Temple Ruins - though you'll have the easiest time hunting them in the Warehouse District of Yusnaan.
Three Star Quests
Bandit's New Weapon
  • Unlocked: Complete For My Child and Heightened Security
  • Quest Item: Organic Carapace x 5, Cactuar Doll x 5
  • Reward: 1300 gil, Electric Guitar, Yellow Propellor

Organic Carapaces are dropped by Gurangatches, which appear most often in the Giant's Sandbox, near the Dead Dunes' Train Station. Cactuars can appear in every area of the Dead Dunes, but they're most common in the small canyon leading to the Oasis Lighthouse.
Desert Cleanup
  • Unlocked: Complete Biologically Speaing and Flower in the Sands
  • Quest Item: Desert Rose x 10, Dried Scale x 30
  • Reward: 600 gil, PSICOM Mobile Medal

Desert Roses are dropped by Earth Eaters in Atomos's Sands, while Dried Scales are dropped by Sahagins in most other sections of the desert. The latter will be easy to kill; the former... well, have fun.
For My Child
  • Unlocked: Third day
  • Quest Item: Cactuar Doll
  • Reward: 500 gil, Cactuar Figurine, Mint Chip Ice Cream

The Cactuar Doll is held by, you guessed it, a Cactuar - or, more accurately, a Cactair. You can find the Cactair in the Dry Floodlands, blocking the path to the Oasis Lighthouse with a sandstorm. Use a Pilgrim's Crux to get at it, then defeat the Cactair to snag the Doll. You can also get Cactuar Dolls from normal Cactuars, also found near the Oasis Lighthouse.