What is the benefit of a single flute, double flute or even triple? The more flutes the better the cut or just the opposite maybe?
Rule of thumb: the more flutes the better the finish but the slower the feed rate.
Single-flutes bits I don't believe are that common, I think they were most seen on narrow straight cutters where there was limited space for multiple carbide inserts.
Double-flute bits are the most common. The great majority of common router bits have two flutes.
More flutes than this in general will give a better cut because the distance around the circumference from cutting edge to cutting edge is reduced. This also spreads the wear more so in theory at least the cutter should stay sharper proportionally longer. Because there is more carbide bonded to the bit they can be substantially more expensive and as mentioned in the rule of thumb they require a slower feed rate.
On older HSS profiling bits more flutes were common, some had six or more. Obviously this was partly to spread the wear since HSS is fairly tough but much less resistant to wear than tungsten carbide, but it was also because they were intended for lower RPMs.