Typical commercial bird baths are usually no deeper than 2″ in the middle, and often taper from shallow edges into the deeper middle. This depth seems to be ideal for the larger birds — grackle, cardinal, etc.
The smaller birds, however, would prefer an inch deep or so. If you observe birds around streams, you will see that they mostly use very shallow, quiet, pebble- or rocky-bottom pools or very slow-moving shallow streams for bathing.
You can make a bird bath more “natural” by placing pieces of slate in the bottom to vary depth, or larger rocks that stick up above the water, but have sloping sides giving birds different depths at which to stand. You can also tip a deeper dish so it is on a slight angle, giving you the gradation of depth.