Any discrete subgroup of finite covolume, whose cusps (i.e., fixed points of parabolic elements) are not the same as those of $PSL(2,{\mathbb Z})$ (i.e., ${\mathbb Q}\cup\left\{\infty\right\}$) must necessarily be not commensurable to $PSL(2,{\mathbb Z})$.

And even if the cusp set is ${\mathbb Q}\cup\left\{\infty\right\}$, the group need not always be commensurable to $PSL(2,{\mathbb Z})$. Examples of this kind have been constructed by Long and Reid in "Pseudomodular Surfaces". Their examples are non-arithmetic, which can be checked by finding elements with $tr(\gamma^2)\not\in{\mathbb Z}$.

Some more examples have been found later by Ayaka.

Any discrete subgroup of finite covolume, whose cusps (i.e., fixed points of parabolic elements) are not the same as those of $PSL(2,{\mathbb Z})$ (i.e., ${\mathbb Q}\cup\left\{\infty\right\}$) must necessarily be not commensurable to $PSL(2,{\mathbb Z})$.

And even if the cusp set is ${\mathbb Q}\cup\left\{\infty\right\}$, the group need not always be commensurable to $PSL(2,{\mathbb Z})$. Examples of this kind have been constructed by Long and Reid in "Pseudomodular Surfaces". Their examples are non-arithmetic, which can be checked by finding elements with $tr(\gamma^2)\not\in{\mathbb Z}$.

Some more examples have been found later by Ayaka Oigo.