How are supernova currently classified?
Of the 4 Supernova observed in our galaxy and the 1 in the Large
Magellan galaxy (i.e. the 1987 supernova) which classification would
each of these fall into?

And Finally 

What is the process which distinguishes each supernova classification?

Regards

Mongolia

As explained here: http://cosmos.swin.edu.au/entries/supernovaclassification/supernovaclassification.html

Supernovae are classified based on the presence or absence of spectral
lines indicating Hydrogen, Helium, and Silicon.  There is a chart and
explanation of the different categories on that site.

Specifically, SN 1987A in the Large Magellanic Cloud is type IIp.

As far as supernova observed in our galaxy, there have been more than
4.  Here is a link listing some historical one (although apparently
not their classifications) http://www.seds.org/messier/more/mw_sn.html