My question has to do with Ravenous trap. Say my opponent casts Thought Scour targeting themselves. Once the spell resolves they have two cards + thought scour itself in the graveyard this turn. I want to prevent them from casting a Tasigur, the Golden Fang/Gurmag Angler. Could I cast Ravenous trap right after thought scour resolves before they get a chance to delve their graveyard?
In order to cast your Trap you must find a way to gain priority between the resolution of Thought Scour and the casting of Tasigur, as APNAP rules state your opponent gets priority (regardless of if the stack is empty or not).
116.3b The active player receives priority after a spell or ability (other than a mana ability) resolves.
However, Tasigur can only be cast at sorcery speed. So if you could get an ability to trigger off of Thought Scour, then your opponent would receive priority with an ability on the stack. When he passes priority to you, you can cast your instant speed Trap**. The easiest way I can think of to cause a trigger would be off of the cantrip. Something like Mind's Eye should do the trick, although I'm sure there are plenty of others.
If you're unable to trigger an ability on the stack off of Thought Scour, then your opponent will be able to delve his graveyard to cast Tasigur.
** - Yes, I passed up the opportunity to say "activate your trap card". Shame on me.
No, you would not be able to prevent the cast. Because it's your opponent's turn, any time a spell resolves, they are given the first opportunity to cast spells (this is called "having priority"). So as soon as Thought Scour resolves, the game "asks" if your opponent has anything to play. In your example, they play Tasigur/Gurmag Angler, and should they choose to delve to pay for the cost, those cards are exiled immediately.
You could still cast Ravenous Trap in response to the Delve card's casting (and the cost would still be reduced), but it would only affect the remaining cards in your opponent's graveyard, and would neither affect the delve nor your opponent's ability to cast the card.