A high frequency noise, or "squeal" from a monitor is usually a resonance problem with one of the circuit components; one (or more) components is vibrating in resonance with an operating frequency used by the monitor's circuitry. It can also be a design defect or the result of running a monitor above its rated frequency specs.
It rarely indicates a soon-to-fail monitor, though the sound can be rather annoying.
TV's have had the same problem for years, and the troubleshooting and repair methods used for TV's are mostly the same used for computer monitors. The most common cause is a loose or ringing flyback transformer. Other coils such as the yoke, convergence coils, vertical output transformer, etc can also produce a high frequency squeal. Leaky or poorly mounted electrolytics may squeal, and in fact, just about any component can be suspect. Other causes may be do to mechanical assembly; two components may be vibrating against one another, or there may be loose or cold-solder joints which are vibrating.
I've been repairing electronic components as a hobby since the mid 1960's, (and later, a BSEE degree), but have only worked on two units with squeal problems. Since I can not hear much above 7500 Hz, I needed the units' owners around to tell me when the problem occurred or was repaired! One unit - a TV - was producing audible (to me) frequencies as well as a high pitch squeal (inaudible to me) emanating from the flyback transformer. Despite all my efforts to fix the existing component, I had to replace the flyback to effect a repair.
Another unit - an IBM monitor - produced an inaudible (to me) squeal, but used to drive my girlfriend crazy. Together, we repaired it - her ears, my labor. I never did determine exactly what the problem was, though I do remember poking and proding a lot of components plus resoldering a good portion of the main circuit board.
Reading the AT forums, it appears that many here are proficient at assembling computers but lack the background required to troubleshoot and repair electronic circuits. For those people, a few suggestions: Try changing the refresh rate, move or tilt the monitor, or even slapping the monitor's case may be a temporary solution to squeal problems.
For those with the knowledge and equipment to repair electronic circuits, some brief guidelines on monitor repair can be found at:
Sci.Electronics.Repair FAQ
One may (or may not) get some helpful info by posting on the Sci.Electronic.Repair newgroup as well.