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.

In this short paper, I sum up the ideas of unparticle field theory (UFT). UFT is a scale-invariant theory. Georgi proposed it in 2007. Some slides (of earlier times) by him are available here on the subject. Despite being a wonderful field theory, it lacks experimental evidence. However, indirect shreds of evidence are possible through the following channels

.

There are more channels other than these. (For technicalities of these interactions, refer to papers.) UFT has a parameter  in the field equations. UFT is unlikely (for now) to be observed because it can be integral values, for instance, 1/2. How would the field theory look? 

In the case of  , UFT is just standard model field theory (these scalar theories are scale-invariant). For instance a UFT with the propagator 

 is phase-space (refer to any paper on UFT).  When , the propagator becomes

which is a familiar propagator in SM. There are many more interesting things about UFT that I cover in the paper with results. You should check first (if new to UFT) papers by Georgi, Tzu-Yiang Yuan, and Kingman Cheung. Georgi is also carrying interesting results in Schwinger's problems. 

Recently, a new term was dropped in this area by Hammer and Son. Unnuclear physics is a non-relativistic theory of unparticles. This EFT claims much more experiments than unparticle physics (which is a relativistic version). A recent talk by Dam Son emphasized the phenomenology of unnuclear physics, which is his next paper (to be released this month). A lot is going in this field. However, it is a newborn field, so a lot can't be said.