  We prove results towards classifying the possible torsion subgroups of
elliptic curves over quadratic fields $\mathbb{Q}(\sqrt{d})$, where $0<d<100$
is a square-free integer, and obtain a complete classification for 49 out of 60
such fields. Over the remaining 11 quadratic fields, we cannot rule out the
possibility of the group $\mathbb{Z}/16\mathbb{Z}$ appearing as a torsion group
of an elliptic curve.
