I have done this repair before (actually twice, once for a broken coupler, once for a slipping clutch), so was surprised when neither seemed to be bad. The only other thing I could think of that was in the drive train for the spin cycle was the basket drive, so I ordered that along with the Cam Agitator Kit as the dogs in it had obviously given up the ghost (probably had been operating without the agitator working for a while; its not one of the things you think about most of the time).
The Basket Drive comes with a new clutch, so even though I had just replaced it a few months ago, I pulled the old one: lo and behold, there was a small bunch of steel wool in it, just enough to get between the plastic clutch material and the metal exterior surface. (No idea how that could have gotten in there.) Obviously, the cause of the non-spin. Replaced the clutch, and then the basket drive, and got everything back together. I always find the hardest part of working on my Whirlpool washer to be putting the cabinet back on - I have trouble getting the sides to line up with the base, but after watching Matt do it so easily on the video, I found that it really wasn't too hard and managed to get it right on my second try.
The videos from Appliance Parts Pros are so great. I usually watch them a couple of times before I start and then freeze them incremently while actually doing the work so I don't forget some little piece; often those little pieces make a big difference.