  For each integer $\ell \geq 1$, we prove an unconditional upper bound on the
size of the $\ell$-torsion subgroup of the class group, which holds for all but
a zero-density set of field extensions of $\mathbb{Q}$ of degree $d$, for any
fixed $d \in \{2,3,4,5\}$ (with the additional restriction in the case $d=4$
that the field be non-$D_4$). For sufficiently large $\ell$ (specified
explicitly), these results are as strong as a previously known bound that is
conditional on GRH. As part of our argument, we develop a probabilistic
"Chebyshev sieve," and give uniform, power-saving error terms for the
asymptotics of quartic (non-$D_4$) and quintic fields with chosen splitting
types at a finite set of primes.
