It depends on the circumstances.
Quite often, small leg-supported tanks or silos just aren't built to any standard. Or they may be built to API-650 or other standards "to the extent applicable".
An atmospheric tank is exempt from ASME Section VIII, but could be built to the code if desired. If the fabricator routinely builds pressure vessels, and if the tank is vessel-shaped (IE, F&D heads), this would probably be pretty reasonable.
I believe UL-142 and possibly API-620 allow tanks to be leg-supported, but don't give any guidance on the design, and include additional testing, etc., that may not be desired.
There is an AWWA D100 standard for water tanks that includes leg-supported tanks. Typically used for large municipal water towers. It gives some requirements for the design, but doesn't include a cookbook-type design, either.