> I searched the building code and found nothing about ensuring power outlets are grounded
The major US building codes leave the electrical to the NEC Code.
BTW, neither BOCA/IBC nor NEC has any legal standing; that's up to the town/city. But the majority of towns formally adopt one of the building codes and an electric code (the NEC is the only widespread electric in the US) with little or no local modification. (NYC, Chicago, Boston, and I hear Austin have local details.)
GroundED and GroundING crept into the NEC over a 50 year span. Even though 250 is about grounds, the full details are spread all over the book.
For recent legacy existing work (built well past 1972), there must be two 20A grounded circuits in kitchen/pantry. Two so that you can plug in a hi-power toaster and a hi-power Mr Coffee without overload. Grounded because before double-insulation or GFIs, grounding was the best bet around water pipes. When GFI costs came down (early 1980s?) then any outlet within reach of plumbing (specific distances in code) shall be GFI.
I don't have an NEC here. I do know that "ALL" recent construction that I have seen is 3-pin outlets everywhere (with odd exceptions like clocks and other dedicated quasi-permanent outlets). And a 3-pin outlet "must" have an effective groundING wire back to the fusebox (or read a special Exemption using GFI to provide life-safety without a useful audio ground). I "assumed" 3-pins are generally required, but I can not cite chapter and verse.
More recent changes want TR (hairpin proof) outlets many places, and Arc-breakers in bedrooms. Neither change affects your studio.
When I had my new-room wiring final-inspected, the inspector had a 3-light tester to sample a majority of outlets for the "OK" indication. I'd think that failing the 3-light test would be a legitimate safety concern.
The Code does not prescribe Good Audio. And landlord or inspector may shrug-off some faults where they do not see a safety issue (and arguing only makes things tougher). For small studios, find one preferably grounded outlet somewhere, safely run a fat 3-pin cable to the studio, and get a many-outlet power strip to plug all your goodies into. As long as they all reference each other, there should be no significant voltages between panels. If they also all reference building ground, there should be no voltage to radiator or sink, and lower EMI from building wiring ambient field.
Codes upon codes. The NFPA (fire) code prohibits power-strips, daisy-chaining, and permanent extension cords. This cost me $4,000. I had 3 wall-outlets in my audio/computer workshop, so it was ALL extension cords and powerstrip into powerstrip. The upshot was I got three new circuits and nine new 4-hole outlet boxes, and it really did work better. But if you are not prepared to invest, then be sure ALL your hay-wire is "temporary", and keep it looking like you just wired-up this morning and will un-plug before you go to bed.