If you aren't doing a DSB, my recommendation (and what I do) is 3/4" to 1". Easy to clean, nice to look at, functional as well as aesthetic.
If you're doing a DSB it needs to be at least 4" (4-6") Anything in between will be more problematic before a DSB is.
I don't recommend DSB - they do need to be maintained, and they do get "full" - ie they do hit a saturation point and need to be changed out or partly changed out every 2-3 years or so.
More work than they are worth, IMO, and they can become rather unsightly when they start working - all that pretty white sand in the beginning, will take on all kinds of interesting colours as the detritus within starts to be processed - there will be layers of funk, some green, some black, etc.
DSB were all the rage about 8-10 years ago. Once people had them for a while and started encountering problems, many got away from that method (and went bare-bottom!)
Personally I like the more natural look of sand - a shallow, well-maintained bed gives you the best of both worlds.
Jenn