>When first powered up it makes a repeating "wheeeip-wheeeip" sound, and
>eventually as it warms up the display momentarily appears just as the
>sound peaks, then disintegrates into a wiggle of Lissajous Curves down
>the centre of the screen.  Finally it locks in and the display steadies.

Actually, having had the experience of repairing quite a few monitors
without schematics...  I would shy away from assuming it has anything 
to do with the power supply itself.  The sound is the switching power
supply *trying* to supply the proper current...   

Someone mentioned the Capacitors... yup.  Sun usually used Sony
monitors,
and I will say in general the high-end brands (Sun, Nec, etc) used much
better quality parts -- probably the main reason a lot of the old Sun
and
SGI monitors and the like have lasted so long.  Good design.  I got a
pile
of old Sampo 17" SVGA monitors at one point...  7 for 7 died from a
capacitor that was rated 140V, 85degrees-C, 3000hrs.  They all died
within about 2 years of purchase.  They nicely had put the Cap right
next to a large heat sink, so it was probably real close to 85-degrees-C
(especially
when they shut the A/C in the building off over weekends) and if you
look at it, 3000hrs isn't all that long when people leave their monitors
on 
24/7.  The cap took out a diode and a $9 transistor when it died. 
Replace the diode and transistor, and put in a 160V, 120-degrees-C,
5000hr cap, they've all kept running (minus one that lost a color
because a resistor went bad).  We had Nec Multisync-2's (14" SVGA -
vintage 1988) that we finally tossed at work just from age and nobody
wanting a 14" anymore, but they still worked fine.

Personally I always shut a monitor off when I'm not using it.  Heat is 
then enemy of electrolytic caps and other components changing value over
the years.  Yes, I've heard the whole "thermal shock from powering it on
and off" thing, but I'm not so sure it outweighs the whole problem of
monitors just being passively cooled by silly vents on the top.

I *do* leave my CPU on 24/7, as it is crunching RC5 (or my fileserver or
webserver.. up all the time).  I *always* shut my monitor off if I'm not
going to be using it for hours (such as, when I go to bed, or leave the
house to go to work).

For schematics...  If you have the manufacturer part # (motorola?) you
can call them.  Or Sam's Photofacts might have them.  

I've found in general for any of the monitors I've fixed, its generally
maybe $10-$15 in parts, its more the labor involved.