Obviously, quests that have multiple ways to solve involving different aspects of gameplay - combat, use of skills, finding a certain item, exploring a certain part of the scenario, etc. Possibly and probably, a combination of these things. And it's okay that some of these quests are based on shades of grey, as long as they are not always the same shades - after all, once in a while something is almost white or black in life. There are good people, and assholes/psychopaths. Eventually you'll find some of them.
That said, there are some kinds of quests I'd like to see. Some of those I've seen in games in the past, but nothing wrong with repeating a structure that worked well, right?
Quests that only appear in places you have visited after you have completed quests in other places. I like the idea of an RPG where the "density" of quests from certain places is somewhat constant, so instead of having 40 quests available right at the start in a certain city, you have 12-15 of them, and the others will gradually appear as you progress in the game doing things on your own or in other places. They do not necessarily form a "chain" as they do not have to be literally linked with the same theme, but are indirect consequences of each other. Let's just say you did a quest where you killed a group of raiders in one place. Well, the raiders actually drove a group of wolves away from the nearby town and now the cattle are being attacked by these wolves. Things like that.
A quest where you have two characters mutually accusing each other of the same thing, but depending on your choices you find different evidence pointing to the guilt of one of them; but on mutually exclusive paths where you could never get 100% of the evidence in a single playtrough so you would have to go with your gut.
If it is a realistic or futuristic game, a quest where you need to visit a place that is apparently connected to the deaths of several people, and when exploring the place things happen that you can never fully explain as natural events but that are subtle enough that maybe there is a non-sobrenatural explanation, but that nothing is really explicit in the end and you are left without an answer.
In fact, I like quests that stay open-ended, and you never have the answer whether what you did was really good or bad thing, or whether it will make a difference in the long run. (And let the consequences stay open even at the end of the game).
Quests where you have to prepare a place to suffer an enemy attack, but you can really influence the place in an interesting way, creating artificial barriers (perhaps destructible) at certain points or setting traps to then face the attack.
Quests where for a specific period of time you control a different group of characters (who may have attributes or equipment derived from the choices you made previously in the game) whose choices you make with them during the quest will interfere with the end of the quest when you take back control of your characters.
Quests where you will compete with another group of characters "equivalent" to yours, being able to interact with that group in several points or even simply kill everyone if it is your choice.
Etc.
