Skogsgurra described the reverse of your question. 60 HZ motors operated at 50 Hz.
50 HZ motors operating at 60 HZ: The speed will increase 20% but in general both motor types could perform quite well except for some extra iron losses easily handled by the extra fan cooling due to speed increase.
For three phase motors it is convenient an increase of voltage proportional to the frequency change (60/50) other way the flux is too weak and the speed-torque curve drops with the squared ratio of the flux reduction.
For instance a 380 Volts 50 HZ motor will work at 380*60/50 = 456 V or the STD 460 V 60 HZ in USA. To extend the life use a conservative same power demand (HP).
For single phase motor a voltage match could be not possible for 220 or 120 VAC, this could weaken the motor too much unless you do a re-winding. Other way, you expect similar performance and recommendations as a 3-phase, the other possible problem is the centrifugal switch cutting the starting circuit at 5/6 of the operating speed. This will not be a problem unless a very high torque demanding load is coupled to the single phase motor.