What can be the advantages of using "normal" trigger mode in an oscilloscope?
The 'normal' mode keeps the last triggered picture.
In 'auto' mode the information disappears. 

Most events I study are single events or burst events. (e.g. 4 writes on an SPI interface). I use 'normal' because I can see what happened many seconds ago.
In fact it is this capability which make a digital scope superior to an analogue one. (Except the very expensive ones which had a phosphor memory).