  In this paper, we explicitly classify the minimal discriminants of all
elliptic curves $E/\mathbb{Q}$ with a non-trivial torsion subgroup. This is
done by considering various parameterized families of elliptic curves with the
property that they parameterize all elliptic curves $E/\mathbb{Q}$ with a
non-trivial torsion point. We follow this by giving admissible change of
variables, which give a global minimal model for $E$. We also provide necessary
and sufficient conditions on the parameters of these families to determine the
primes at which $E$ has additive reduction. In addition, we use these
parameterized families to give new proofs of results due to Frey and
Flexor-Oesterl\'{e} pertaining to the primes at which an elliptic curve over a
number field $K$ with a non-trivial $K$-torsion point can have additive
reduction.
