Among some ways of proving the criticality, the most famous is by using $j_{\mu\nu}$ (Lorentz elements) - A method involving a very after-canonical quantization process of deriving critical dimensions can be found in Polchinski's volume. You can check if the action you have written for your strings (Polyakov action) is Poincare invariant. If they are Poincare invariant, then they are in critical dimensions. For bosonic string theory, it is $D=26$ and $D=10$ for superstring. One can find the critical dimensions for superstring by adding fermions using RNS. For M-theory, a close but not similar process can be carried out. If the theory is critical, then we should not fear the super-conformal ghosts that appear as central charges in algebras of string theory. In $D=26$, one can have the vacuum state as a tachyon, which is negative mass squared. In superstring theory, tachyons don't appear. Tachyons are unstable.

We can't talk much about non-critical theory, however, some models show good significance, for instance, $D \geq 4$, but too premature. Also, T-duality is only applicable for critical dimensions in super-string theory (I haven't encountered any support for non-critical dimensions for T-duality). Some good studies are holding for non-critical string theory with its application to AdS (not aware of recent development).

Edit: Paper by Polyakov also had a solution to the critical dimension from 1981 (the original subject of paper was on summation of random surfaces). And, there are also prospects of Liouville thoeries in non-critical dimensions.