They're common on the Slime Island.
Blue Slimes have 3 different sizes; big, medium and small.
The big slime has 24 HP, and breaks up to the medium version when killed. The medium slime has 12 HP, and breaks up to the small version when killed. The small version has 4 HP, and is still aggressive, but seems to do no damage to a player.
You can find Blue Slimes on Floating islands; these are not hard to find when using Rei's Minimap. You will also find them below the islands, because they don't take fall damage when they fall off the island. Rarely, you can find them in Tower dungeons. They will also spawn from the King Slime.
NB: They can also carry other mobs (usually skeletons), making them an even harder enemy to defeat. When a slime is broken into a smaller version, the smaller version may spawn as an Infernal Mob.
Blue Slimes take only half damage from arrows.
Bug: Infernal Blue Slimes frequently will not display the particle effects nor display their infernal name and list of infernal buffs.