What is the ideal depth for a bird bath?
You can find a wide variety of birdbaths to purchase. From the bird’s perspective they are most interested in gently sloping sides, for easy puddling especially for small species and a rough surface to provide safe footing. Ideally the bath should be about 3” deep so large birds can soak and hold a flock that may bathe together. A large stone can be added to create an island. Be sure to clean bird baths often to keep water fresh and clean. (This is important, too, in the summer when mosquitoes lay eggs in standing water). Place your birdbath near trees or shrubs where a bird can perch to dry and preen their feathers. Be sure to offer water all year long, as it's essential for their diet.
A great way to have birds such as robins stick around all winter is to supply a heater or a completely heated birdbath to your yard. Birds have a difficult time melting snow in their beaks and water is as critical to their survival as food. If you're nervous about birds bathing in a bird bath when it's bitter cold (they rarely do) you can always take some metal poultry/garden fencing, cut it to fit around the dish of the birdbath and then crimp the edges. That way birds can dip their beaks in for a drink but won't be able to splash around for a bath.