When does a Mordell curve have non-trivial torsion?

Is there a known simple criteria for when a Mordell curve has non-trivial torsion?

A comment in this question: Family of elliptic curves with trivial torsion
Suggests that $$y^2 = x^3 + k$$ has nontrivial torsion when k is 1, a square, or a cube.

If this is true, it would be nice to see it explained or point to a reference I can read more.

Here is a further reference, extending the result you have linked.

Theorem: If $k$ is square-free and not equal to $1$, the elliptic curve $y^2 = x^3+k$ has no rational torsion points.

Proof: See A. Knapp, Elliptic Curves, Princeton Univ. Press, 1992, Theorem $5.3$.

More generally, there is indeed a classification result as follows.

Theorem: Let $k=m^6 \cdot k_0$, where $m, k_0\in \Bbb Z$ and $k_0$ is free of sixth power prime factors. Then the torsion subgroup of $E\colon y^2=x^3+k$ over $\Bbb Q$ is given as follows: $$ E_{\rm tors}(\Bbb Q)= \begin{cases} \Bbb Z/6\Bbb Z & \text{ if } k_0= 1, \\ \Bbb Z/3\Bbb Z & \text{ if $k_0$ is a square different from $1$}, \\ \Bbb Z/3\Bbb Z & \text{ if $k_0=−432$}, \\ \Bbb Z/2\Bbb Z & \text{ if $k_0$ is a cube different from $1$}, \\ \{\mathcal O\} & \text{ otherwise.} \end{cases} $$