Mayflies typically have plate-like gills that are located on the side of their abdomen. They will have 6 jointed legs that end in 1 claw. Most mayfly nymphs have 3 tails.
To learn more, go to The Mayfly Keys

Stoneflies have no gills on their abdomen. If gills are present they will be found on the underside of the head and thorax. Stoneflies have 6 legs that end in 2 claws. Stoneflies will have 2 short tails.
Go to stonefly adult.

Dragonflies have no tails. A dragonflies body is squat and stout. They have large mouth parts.
Go to dragonfly adult.