  Let $x,h$ and $Q$ be three parameters. We show that, for most moduli $q\le Q$
and for most positive real numbers $y\le x$, every reduced arithmetic
progression $a\mod q$ has approximately the expected number of primes $p$ from
the interval $(y,y+h]$, provided that $h>x^{1/6+\epsilon}$ and $Q$ satisfies
appropriate bounds in terms of $h$ and $x$. Moreover, we prove that, for most
moduli $q\le Q$ and for most positive real numbers $y\le x$, there is at least
one prime $p\in(y,y+h]$ lying in every reduced arithmetic progression $a\mod
q$, provided that $1\le Q^2\le h/x^{1/15+\epsilon}$.
