I don't actually own a flat-panel LCD display for my computer, but I've been looking at what's available because people occasionally ask me about them, after I wrote an article on the subject (based on my journal posting) late last year.
Things are now at the point where I would recommend a flat-panel display over a traditional picture tube for almost anyone who doesn't regularly require different screen resolutions or isn't doing super–colour-critical work.
Things always get cheaper, of course. These days I'd recommend spending $600-750 (Canadian) for a 17" flat-panel display with 1280x1024 or 1280x960 pixel resolution, rather than a 15" at 1024x768 (and $350-500).
A 20" or bigger LCD screen with 1600x1200 resolution would of course be lovely, but that's getting into the $1000+ range. If you're spending that kind of money, you have probably done more research than I have. (As I said, I still use old-style picture tubes myself right now.)
Hitachi, Iiyama, Samsung, Sony, NEC-Mitsubishi, Proview, ViewSonic, Planar, Sharp, and Apple (only for Macs) all make top-quality displays, but most also have a range of models with varying quality and features. Try to examine the exact display you will be buying, so that you can look for dead pixels, a problem unique to LCDs.
Chances are your current computer will plug into an "analog" or "analog/digital" LCD, but "digital-only" displays might require a new video card, so that's worth keeping in mind.
If you have a limited budget, 15" models can be had in the $350-500 range (Canadian), maybe less if you're lucky. But don't shop rock-bottom; your eyes will suffer accordingly. If you don't mind the heat, power consumption, and loss of desk space, it would be better to spend less money on a traditional 17" CRT (picture tube) if that's all you can afford.
As for simple recommendations for particular current models, the Samsung SyncMaster 171N seems to be a good 17" model right now, according to PC Magazine, as does the Sharp LL-T17A3. Their recommendations for 15" panels (with the Planar PL150M and Sharp LL-T15G3 coming out on top) are here.
That gives you four good choices to examine.