For example, how small should peak widths be, and is there a gauge for s/n ratio? Are there any other characteristics to look out for that is specific to DSC curves?
Additionally, how can I deduce whether supercooling has occured?
If you mean the parameters related to performace of a particular DSC, then signal-to-noise ratio (in both isothermal and linear heating), dynamic range (i.e., the maximum measurable power), overall heat transfer coefficient, baseline stability etc.
If you mean how to describe a DSC curve using peak height, width (or FWHM), peak temperature and similar characteristics: such parameters strongly depend on many experimental conditions such as DSC type and construction (heat-flux/compensation, furnace geometry), sample mass and geometry, heating rate etc. This dependence renders these parameters almost useless. Peak area - the enthalpy change is an exception as it is a function of state.
However, the aswer also depends on what kind of process are you investigating (phase transition - 1st/2nd order, reaction, ...)
I agree with Dr. Dubaj, and would add an example or two.
For DSC, the heating rate is an important parameter. Faster heating rate will increase the signal, thus making the method more "sensitive" to picking up or displaying thermal events. On the other hand, however, faster heating rates lead to broader peaks and less resolution with respect to the temperature associated with events, as well as perhaps phenomena such as enthalpic overshoots, etc. All of these affect the shapes of curves, as well as what is sometimes detectable, etc. Thus, it's important to understand what you are looking for and optimize your procedure, which sometimes occurs by looking at your results, modifying heating rates, etc.
Regarding supercooling, one way to do this is to take a solid and melt it, then compare the thermogram when cooling the melt. There should be a clear melting peak (for instance, for a pure crystalline material) when heating. If there is no crystallization peak corresponding to the heat release when cooling, you can infer supercooling. Of course, this comes with the caveat that cooling and melting are completely different kinetic events, even if they are theoretically similar from an equilibrium thermodynamic standpoint. (Crystallization is slower and more complex, and there is no guarantee that cooling will produce the same crystalline form as what you started with.)
As you can tell, these are more complicated than running the instrument and reading data... just some points to consider.
as newcomer in the field, i would recommend one of these courses as starting point: check out mt.com/ta-webinars and pick up a pre-recorded webinar. Maybe DSC basics and applications is the right one to begin with.