My answer may be considered wrong by some of you guys, but.... this is what works for me. Or at least, this is how I like to do it.
Fine sand to over 1000 grit, I like 1500. I prefer to sand by hand when I get beyond 400 grit. I clip a clamp on light to my desk top so the light source is behind the piece I am sanding so I can see any scratches and areas that make look a little duller than the rest of the wood.
For the hand sanding I use the stick on round sandpaper that is intended for random orbital sanders. I cut the piece into 1/4s. Then I peel off the backing and stick the sandpaper to my 1st 2 fingers. This way the sandpaper doesn't slip around when I am sanding.
I blow off the piece in between grits to insure there is no abrasive left from previous grit paper. I feel if you sand to the point where it looks finished to you that is fine enough.
(a quick note) Do not apply oil in a cold or damp shop! Make sure the wood and the oil are both at a room temperature above 50f.
After blowing off the piece then get out some oil. Danish, tung, tru or teak.
I take an old t-shirt and get a little oil on the cloth and rub it in with my fingertip. Let it set about 30 minutes and rub down with dry t-shirt. About an hour later I wipe on more oil and repeat. 2 or 3 times should be enough. I just make sure everything looks even under a light.
After it sits for a day or 2 in a heated room you can apply some wood paste wax. Pretend like you are shining a pair of shoes. Those of you who have been in the military know what I am talking about. Buff by hand with an old towel or whatever cloth you have on hand.
The stabilized wood usually looks good enough after just fine sanding, but using just a touch of oil makes the colors look more vibrant and the figure flashier in my opinion.
The hand sanding and oil might be overkill, but for me it is relaxing. Allows me to wind down and relax.
Disclaimer:
You might have a better way to finish stabilized wood. This is just how I like to do it.