Distribution of prime numbers modulo $4$ Are primes equally likely to be equivalent to $1$ or $3$ modulo $4,$ or is there a skew in one direction? 
That is my specific question, but I would be interested to know if there exists a trend more generally, say for modulo any even.

In general, if $\gcd(a,b) = 1$, the number of primes which are of the form $b$ modulo $a$ is asymptotic to $\dfrac{\pi(x)}{\varphi(a)}$ where $\pi(x)$ is the number of primes $\le x$. As you see the asymptotic formula is independent of $b$ hence for a given modulo all residues occur equally in the long run.
