  We examine canonical bases for weakly holomorphic modular forms of weight $0$
and level $p = 2, 3, 5, 7, 13$ with poles only at the cusp at $\infty$. We show
that many of the Fourier coefficients for elements of these canonical bases are
divisible by high powers of $p$, extending results of the first author and
Andersen. Additionally, we prove similar congruences for elements of a
canonical basis for the space of modular functions of level $4$, and give
congruences modulo arbitrary primes for coefficients of such modular functions
in levels 1, 2, 3, 4, 5, 7, and 13.
