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You can fairly quickly determine if a transformer is grossly bad. This does not say anything about the subtleties of it's tone, only that it is functioning. Tests of relative tonal "goodness" are also possible, but require a lot of equipment and experience to do correctly. For the tests described here, you'll need an ohmmeter for the simplest testing, and for more advanced tests, a means of measuring AC voltage and current simultaneously, such as a pair of VOMs or DMMs, and a 110/120 to 6.3VCT filament transformer, and either a variac (variable transformer) or a light bulb socket in series with the primary of the filament transformer to limit the power you put into the transformer under test.
If you do the advanced tests, be aware:
Both the filament transformer and the transformer under test will have at least AC line voltage on them, an may well have much higher voltage, several hundred volts on one or more windings. You are therefore in danger of being KILLED if you are not both knowledgeable and careful about how you do these tests. These tests are clearly hazardous as defined in the warnings section of this page.
Unplug the amp and drain the capacitors before you start. Do not remove the transformer from the amplifier before you start. Remove the output and rectifier tubes.
Identify which wires are which by circuit connection. The primary will have two (single ended), three (push pull, or very rarely single ended with ultra linear tap), or five (push pull with ultralinear taps) leads. Noting which leads connect to the B+ line, measure the resistance from the B+ lead to each plate of the output tube(s). Write the resistance reading down. Measure the other side if it's puah pull. Plate windings are almost always in the tens to a few hundred ohms of resistance. A reading much over 1K is pretty sure sign that the winding is burned open. If it's open, the transformer is dead. [There are ways to fix "slightly-dead" transformers, but this is extremely risky, and probably doomed to failure. I'm not going to write down how. Don't waste your time.]
While you're at it, measure the resistance of every lead to the chassis. This reading should be very high, hundreds of K or preferably megohms. A low reading here indicates a short to the transformer core - again, dead transformer.
Do a similar test on the secondaries - look for open windings, or shorts to the core. Finally, test for the resistance from primary to secondary. A low value, under a few K indicates an internal short, and a dead transformer.
If the transformer fails any of the simple test, it's dead. Replace it. If it passes all of the simple tests, it may still have an internal short Sometimes a primary will have enough resistance that the transformer does not burn up or burn the wires in two , but will just bog down, have low power and sound bad. If you suspect that is the case and you don't have the skills to do the advanced tests, take it to a competent tech.
Do the shorts/opens tests above first.
Identify a secondary winding, 8 ohm tap if it's available, and hook up one and only one winding to either 1/2 of the 6.3VCT or to the variac. Make D^&%ED sure the variac is all the way down if you're using one of those. Make sure that no other leads are connected (or shorted together, or touching your screwdriver on your bench or... well, you get the idea). Check for safety load resistors on the output jacks or secondary windings. These can cause you to erroneously think it is bad. There must be no loads on any winding. Put your voltmeter on the winding, and the current meter to measure the AC current through it, hook up the 3.15 VAC source, and turn on the AC - not to the amp, but to the AC feeding the winding you're driving. The voltmeter should measure 3.15 (or close) volts AC, the light bulb (if used) should NOT be lit brightly, and nothing should be humming or smoking ;-). There should be only a little current going through the winding. If the voltage is lower than 3 volts, or you are pulling amps of current, then there is a load on the transformer, internally since you have disconnected all the leads, meaning that there is an internal short. (Which means that you ought to be sure that there is no other load on the transformer, because you're going to throw it away if it has an internal short - be sure you're right!)
If all is well, no smoke, flames, loud hum, poultergeists, or lightning, measure the voltage that now appears on the other windings. The voltages will be equal to the ratios of the voltages that will appear on these windings in normal operations. For the primaries of output tubes, this can be up to a couple of hundred volts, so don't think that you can relax your safety precautions - it can still kill you, even driven with 3VAC on the secondary of the output transformer. The half-primaries of the output transformer should have identical voltages on them. Secondaries should have multiples of the voltage you're putting on the tap you chose. If you used an 8 ohm tap, a 4 ohm tap will have about 2.3VAC on it, a 2 ohm tap will have 1.6VAC on it, and a 16 ohm tap will have about 4.5VAC on it.
If the transformer passes all these tests, it's almost certainly good.