I don't have a screenshot, but I know that some daily quests are the following:
1.) Completing a town's secret/extra dungeon. My Tein in his mid-30s had to complete 1-S twice.
2.) Normal enemy kills. (Usually with a stupid number. Ex: 283)
3.) Dismantling equipment.
4.) Gear-tuning equipment. (I'm pretty sure this is what Lunarix's 5th quest is.)
5.) Running any dungeon with a party.
6.) Using Finishing Moves
7.) Completing any dungeon(?) (They probably have to be around your level.)