The left picture shows my standard grip - 3 fingers on the barrel (the 4th finger is not touching, although it might look like this on the picture). The small finger is spread away slightly. 

The right picture shows a 4 finger grip on the same dart. 

Both pictures show my thumb placed well behind the dart's center of gravity -- good for touch. The tip of the dart of course points nicely up. 

(the dart used is a Bottelsen Hammer Head "Bernd Hebecker", 18gr barrel) 
 

These 2 pictures show that it is often no problem to use the same grip on a different dart. 3 fingers touch the barrel which is more ton-shaped and shorter -- 4 fingers wouldn't fit on the dart here. 

(the dart is Harrows "Dennis Priestley")

Both pictures show the "pencil" grip from different angles. This grip is widespread and good to use, at least good enough for world champ Phil Taylor. 

(the dart is again my "Bernd Hebecker" Hammer Head) 
 

A grip used by one of my teammates. The left picture shows the "original", the right me trying to copy it on my dart. This grip is a bit on the extreme side, so not recommended for beginners.