© 2017 Mathematical Sciences Publishers. For each integer ℓ ≥ 1, we prove an unconditional upper bound on the size of the ℓ-torsion subgroup of the class group, which holds for all but a zerodensity set of field extensions of Q of degree d, for any fixed d ε {2; 3; 4; 5} (with the additional restriction in the case d D 4 that the field be non-D 4 ). For sufficiently large ℓ (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.