When using a bike on a road in the USA, is it ever legal not to have separate brakes on the front and back wheels?
Many places have adopted the Uniform Vehicle Code language. It currently states:
12-706. Brake required Every bicycle shall be equipped with a brake or brakes which will enable its driver to stop the bicycle within 25 feet from a speed of 10 miles per hour on dry, level, clean pavement. (FORMER 11-1207(c)).The goal of the UVC is to have consistent laws between states, counties, and municipalities. It doesn't entirely work - often the process of changing the laws is too complicated, so many outdated variations exist.
In 2006, Portland, Oregon saw an interesting challenge a similarly-worded law. A cyclist was cited for failure to have two brakes, even though the fixed gear on their bicycle allowed them to meet the requirements of the statute. (Ridiculously, the previous UVC phrasing that the ORS statute is based on doesn't even require the ability to stop, only the ability to skid.) The judge and ticketing officer's contention was that a brake must be a separate device.
Meeting the UVC phrasing is your best bet for a US-wide policy, but definitely review the local statutes for the places you will be riding if you are concerned.
I don't recall anything in the Ohio Revised Code that requires bikes to have any brakes. I've never seen a single speed bike around here that had anything other than the standard rear coaster brake. While that doesn't prove legality, the shops around here tend to be really picky about all the details (such as not wanting to sell/service bikes without sound making devices back when those were required).
It depends on the state, and some states have bizarre wordings.
For example in Minnesota from https://www.revisor.mn.gov/statutes/?id=169.222
(b) No person shall operate a bicycle unless it is equipped with a brake which will enable the operator to make the braked wheels skid on dry, level, clean pavement.
It's kind of hard to tell if this means one or two brakes is required. However, it is enforced that a single front brake is acceptable, yet skidding the front wheel is never a good thing, but not doing so doesn't mean the brake is not capable of safely stopping the bicycle.
The laws may differ from county to county and state to state, but as it stands for California:
V C Section 21201 Equipment Requirements
(a) No person shall operate a bicycle on a roadway unless it is equipped with a brake which will enable the operator to make one braked wheel skid on dry, level, clean pavement.
By this interpretation, so long as you have a brake that allows you to skid your tires, it is legal for you to ride your bicycle. Some may even argue that riding fixed gear and skidding the rear tire technically makes for a legal brake equivalent.
I do however feel that this law is lacking, as it is near impossible to skid the front tire with a brake on a level, clean pavement, and a disgruntled officer may use it as an excuse to ticket you, even though it holds 80% of the braking power. But that's another tangent...