The first test, which is not 100% but is a good start:
Assuming you are using a DMM with a "diode" range,
1 Put the test meter positive lead to the base of the transistor and then measure to the emitter and then to the collector.
2 Put the test meter negative lead to the base of the transistor and then measure to the emitter and then to the collector.
With the leads one way you should measure about 0.7v, them the other way round you should measure an open circuit.
This simple test will usually find open or short circuit junctions in bipolar transistors (not FETS or MOSFETS), but is easily confused by the rest of the circuit still connected to the transistor.
Dont forget that the power to the circuit board must be turned off.
JimB