The advantages are simplicity (one of everything), narrowness (the bike is one cylinder wide), steady power delivery, lightness. For these reasons most dirt bikes are singles. On a cruiser it makes the engine look bigger.
The main reasons people are attracted to singles are emotive issues associated with the noise and vibration. Singles also have, of course, a very long heritage in the world of motorcycling.
Singles have a technical ceiling on their performance because of the piston speed (the actual velocity of the piston itself, which has to travel in a long sweep up-and-down compared with the two pistons on a twin, each travelling half as far). The other limitation is the burn time, how long it actually takes the fuel air mix to burn: the flame front has to go from the spark plug to the edge of the cylinder, and this is a greater physical distance on a single.