Do we always use Sophie-Germain primes $1\bmod 4$? A prime $p$ is Sophie Germain if $2p+1$ is also prime. Wiki says if $p\equiv3\bmod4$ then $2p+1|2^p-1$. This seems to put a huge restriction on density of Sophie-Germain prime if they are $3\bmod4$. For this reason do all systems use Sophie-Germain primes with $1\bmod 4$ for discrete logarithm?
This seems to put a huge restriction on density of Sophie-Germain prime if they are $3 \bmod 4$.


I'm not sure how you come to that conclusion; whether $2p+1$ is a factor of $2^p-1$ would not have any immediately obvious consequence to $2p+1$ being prime.


  For this reason do all systems use Sophie-Germain primes with $1 \bmod 4$ for discrete logarithm?


Actually, Sophie-Germain primes of the form $3 \bmod 4$ would appear to be more prevalent; for example, the primes listed here are all of the form $3 \bmod 4$.  One reason is that it makes the value $g=2$ be a quadratic residue, that is, its use during the DH operation doesn't leak the lsbit of the private exponent, and that's appreciated by some people.