Looking for supernovae from the back garden
A supernova is a star tearing itself apart, bright enough to stand out against the whole galaxy that hosts it. Amateur observers have found them for as long as the equipment has allowed, and they still do, because the sky is large, the events are unpredictable, and patient coverage of many galaxies is exactly the kind of work that professional surveys cannot do everywhere at once.
What you are looking for
Two broad kinds turn up. One comes from a white dwarf in a close binary reaching the point where it detonates, and these are prized because their peak brightness is consistent enough to serve as a distance measure. The other comes from the collapse of a massive star's core, and these are messier and more varied. To a patrol observer both look the same at first: a point of light in a galaxy that was not there last time.
Build a target list and stick to it
Success comes from a list, not from wandering. Choose galaxies that are bright, reasonably large in your field, and well placed from your site through the season, and revisit the same ones on the same framing every clear night. Consistent framing is the whole trick, because comparison is far faster when the new image lies on top of the old one. Keep the list short enough that you can cover it in a session, then extend it as the routine becomes automatic.
Comparison is the skill
The work is in the comparing. Blink the new frame against your own reference frame of the same galaxy, at the same scale and orientation, and train your eye on the spiral arms and the outskirts where a new star is easiest to miss against structure. A new point that sits on a knot of star formation is exactly where you should expect one, and exactly where the eye slides past it.
Rule out the ordinary first
Most candidates are not supernovae. Asteroids move, so a second image later the same night settles that. Cosmic ray hits and hot pixels appear on one frame only, so nothing counts until it shows on two separate exposures. Known variable stars, cataloged novae and internal reflections near a bright star all produce convincing false alarms, so check a catalogue and check the same area of previous images. Only when the candidate survives a second exposure, a second night if possible, and a look at the archive does it deserve a report.
Reporting, and why it matters quickly
A confirmed candidate is reported to the transient naming service that professional and amateur observers share, with the position measured as accurately as your images allow, the brightness estimate, the instrument and the exposure details. Speed matters, because the interesting spectra are the early ones: a discovery reported promptly can be spectroscopically classified while the star is still rising, which is what turns a point of light into science.
Equipment and expectations
A modest telescope with a sensitive camera, a stable mount and reliable pointing beats a large telescope used occasionally. Light pollution hurts but does not stop you, because you are looking for a point source against a galaxy, not faint nebulosity. Expect to find nothing for a long time. The observers who eventually succeed are the ones who kept the routine going on ordinary nights, and they will tell you that the routine itself is most of the pleasure.