When a three phase motor is "single phased", it is a power system problem, not a motor problem. A three phase motor needs three EQUAL phases in order to operate properly. When the symmetry of the motor is interrupted by the loss of a phase, the motor will die quickly unless the controls have single phase protection. Many heater type overload relays do not have this.
Single phasing occurs as a result of several possibilities. A loose wire, a bad connection, bad starter contacts, overload relay problems, a bad breaker, a blown fuse, and other things can cause this destructive condition. Obvious signs are a louder than normal humming from the motor and/or a shaft that vibrates rather than rotating.
Testing for this possibility needs to be done quickly since motors are not happy with this condition at all. The obvious test is to look at the current in each phase. This is where multiple meters will help so you can see all three phases at once. You can also look at the voltage, again with multiple meters if possible. I look at the phase to ground readings first. The phase to ground voltage will equal the phase to phase voltage divided by 1.7; thus 480 volts phase to phase will be 277 volts phase to ground. The advantage of taking phase to ground measurements is that each reading is independent of whatever is happening in the other phases. However, you can read phase to phase if you want. You would see an unbalance there too. The phase to ground reading will show you the bad phase, though; this will make troubleshooting easier.
These tests need to be made as close to the motor as possible, preferably in the motor's connection box while the motor is driving the load. If the motor is not connected, or you take your readings at the starter or breaker with the motor off, you can get fooled. A bad set of contacts in a contactor or breaker can just barely touch and still tell you that you have good voltage. Ask those same contacts to deliver enough current to run a loaded motor, and the voltage will take a dive.
You could continue to test at various stages of the power system upstream of the motor, but that keeps subjecting the motor to the stress of running in the single phase condition. Otherwise, make sure the circuit is off and locked out, and then start taking things apart.
The first place to look is at those suspect contactor contacts. But, Bo Diddley said "you can't judge a book by looking at the cover." Contacts can be like me - real ugly but still functional. Contacts that are gone don't work very well, though. Also look at the connections in and out of the contactor. Loose or burned wires or terminals are probably the second most frequent offenders.
If the contactor looks good, take continuity readings from the line to the load side of each phase of the overload relay. It should look like a short circuit. A word about overload relays here. With today's motors being smaller than their U-Frame predcessors, you need a fast overload relay with single phase protection. An IEC Class 10 thermal overload relay works very well. You can get more expensive, solid state models, but the Class 10 thermal relays work well for all but the most sensitive applications. Be careful with replaceable element versions, though. They are usually Class 30 (slower) and don't have single phase protection.
If everything is good at the starter, check wires, connections, and devices ahead of the starter until the problem is found and corrected. Once you have three good phases again, you should see a voltage balance of within two or three percent. Your motor will be happier, healthier, and have a shiny coat.