Flybacks fail in several ways:
More than one of these may apply in any given case.
First, perform a careful visual inspection with power off. Look for cracks, bulging or melted plastic, and discoloration, Look for bad solder connections at the pins of the flyback as well. If the TV or monitor can be powered safely, check for arcing or corona around the flyback and in its vicinity,
Next, perform ohmmeter tests for obvious short circuits between windings, much reduced winding resistances, and open windings.
For the low voltage windings, service manuals may provide the expected DC resistance (Sams' PhotoFact, for example). Sometimes, this will change enough to be detected - if you have an ohmmeter with a low enough scale. These are usually a fraction of an ohm. It is difficult or impossible to measure the DC resistance of the HV winding since the rectifiers are usually built in. The value is not published either.
Caution: make sure you have the TV or monitor unplugged and confirm that the main filter capacitor is discharged before touching anything! If you are going to remove or touch the CRT HV, focus, or screen wires, discharge the HV first using a well insulated high value resistor (e.g., several M ohms, 5 W) to the CRT ground strap (NOT signal ground. See the section: Safe discharging of capacitors in TVs and video monitors.
Partially short circuited windings (perhaps, just a couple of turns) and sometimes shorts in the focus/screen divider will drastically lower the Q and increase the load the flyback puts on its driving source with no outputs connected. Commercial flyback testers measure the Q by monitoring the decay time of a resonant circuit formed by a capacitor and a winding on the flyback under test after it is excited by a pulse waveform. It is possible to easily construct testers that perform a well. See the companion document Testing of Flyback (LOPT) Transformers for further information.