Near preimages, applicable to Bitcoin? Bitcoin mining relies on generating a smaller hash than the so-called target (a function of the so-called difficultly), thus is vulnerable to a truncated preimage attack (you just need to obtain a certain number of leading zeroes).

I believe this is called a near preimage attack.

What is the current status of attacking truncated second preimages of SHA-256?
The current status as of the time I write this is: There are no known attacks on second pre-images for truncated SHA-256 that are faster than brute force.