I have an '04 LS V8.
I absolutely love mine. It looks great, and handles very well while still being roomy.
I typically get 22-24 mpg, according to the computer, with highway and highway-like driving.
I've had mine for just over two months now(about 1500 miles), and haven't had any mechanical trouble at all.
I'm told that the 00-02 models had some teething problems. For that reason, I avoided them when I was looking for one to buy. The 03 and up models are supposedly more reliable. Like I said, though, I really haven't had mine long enough to have any problems.
There are some great bargains to be had on these. The '06s seem to be holding their value pretty well, but the '04s and '05s are selling for anywhere from $10,000-15,000. Not too bad considering that these were $40,000+ cars when new not too long ago.
I personally ruled out the V6 early in my search, because the rated gas mileage isn't much better(about 1 mpg as I recall), and I found the greater enjoyment of the V8 well worth it. Apparently, most others felt the same way, as the V6s are relatively hard to find-in fact, they discontinued the V6 version for '06.
One thing that I should caution you about-the LS does not ride like a Lincoln. The suspension is a lot stiffer and tuned to better handling, not the smooth, cushy ride that you'll get from a Continental or Towncar(my Mom has a Towncar, and my dad a Continental, so I'm fairly familiar with them).
I can honestly say that I wouldn't trade my LS for anything.