Nintendo has released a Splatoon demo that you can download from the eShop. It's more like a beta test than a demo because it's only playable during certain times.
The first (and only) announced session is for three hours on Saturday, May 9th.
The demo lets you play in a 4v4 Turf War match:
Rather than three straight hours, Nintendo is activating the game for an hour at 4am, an hour at 12pm, and an hour at 8pm. That means most of you will be able to play for at least one of those slots, while the dedicated among you can play in all three.
If you plan on buying Splatoon you should probably download the demo even if you can't play because it gives you a 10 per cent discount on the full game.