There are ways to test transistors, using special transistor testing equipment. You have to remove the transistor from the circuit then test using this device, and while it does give you an accurate diagnostic of the transistor, it's a very slow process.  In some cases this is necessary when dealing with more complex problems.

Many times a quick check will be accurate enough to point you in the right direction. A quick check will tell you if a transistor is bad, but it will not tell you it's good. If a transistor you are testing passes the quick check you must keep in mind it may still be bad. This fact should not stop you from moving on to another logical place to look for your fault, but there is always that chance you will have to back up to the original location.

Normally transistors fail by shorting not by opening up. With the power unplugged and the transistor in circuit, set your meter to 'diode check.'  Pick any two of the three legs of the transistor and connect your meter to it. If the reading is infinite or around .4, this is a good sign.

Now, pick another pin and place your probe on it. If you get the same reading as above

you are still in good shape (either infinite, .4, or slightly higher).  Keep doing this until you have tried every combination of probe and pins. If your meter drops to .001, .002, or any reading similar to this you have a shorted transistor.  Please keep reading because there are a few caveats to keep in mind.

If the transistor is connected across a transformer, like a flyback, one set of your readings will fall close to zero, this is because you are across a coil and it will fool you into thinking the transistor is bad. Simply pick another set of pins and check for an infinite or .4 reading. Normally when these transistors short they will short in all directions, so you will not get an infinite or .4 reading from any leg.

You also need to watch out for capacitors in the circuit. Initially, when you touch your probe to the pins of the transistor, it may give you a reading that will make you think it is bad, but keep your probes on the transistor and watch your meter to see if the numbers start to climb. If the numbers are climbing you have a capacitor in circuit and your transistor is not shorted.

A few types of transistors will not test correctly, like Darlington’s and some FETs.

It only takes a couple of seconds to perform this test, and once you get the hang of quick checking transistors, you will cut down the time it takes to pinpoint your problem.