Strength of block ciphers under attack [by government level attackers] So lets say the government has set their top priority to cracking a Veracrypt or Lukes or whatever volume containing highly confidential information.
It uses AES or AES(Twofish). How would they start an attack and how efficient would it be?

Ps: If most block ciphers really are so vulnerable to side channel attack why are most governments still using them for classified information?
Obligatory XKCD: coercion (or/and bribery) works. Other usual strategies are


Rigging the particular machine used by the data owner.
Otherwise spying key-presses on that machine to obtain the password.
Password search (including by compromising this user's other passwords and trying variations); that's an industry that has several professional-looking offers, targeting at least law-enforcement budgets (your guess of other users is as good as mine). It works for encrypted data at rest, as in the question.
Finding the desired data (or the password) in a non-encrypted area, such as swap space, sidestepping the question's problem.


Directly breaking AES (even AES-128) by brute force is so hopelessly hard that it is unreasonable to even attempt that.

Side channel attacks (like timing attack and DPA) are entirely useless for encrypted data at rest, because the key is not there to be extracted.

AES implemented with AES-NI or other hardware is immune to timing attack, and I believe is immune to DPA on today's mainstream CPUs, except perhaps with most direct access to the CPU using the key (as feasible in Smart Cards); that's a risk, but a manageable one.