Lately, when I drive my car at speeds over about 60 or 65 mph, for just a few moments, it will feel like the back end is fishtailing. Other times, it feels like I’m getting hit with really strong wind, even when there isn’t any.
It’s a '91 Toyota Celica, in overall pretty good condition.
Diagnose me 
Well, alignment and tire pressure should be checked. An allignment test might reveal damage to suspension links, arms or stabilizer bars.
How many miles on this puppy? It could be worn suspension parts like shocks/struts. These parts when worn will give you that feeling, esp as speed builds.
Tire/wheel vibration tends to come on right around the mileage mentioned. With a front tire, you can feel/see the steering wheel react. With a rear tire, it sometimes gives a “weird feeling,” possibly including what you’ve mentioned.
At the top of my suspect list is a tire. Out of round, out of balance, plies separating, whatever – it sounds like a tire is unstable.
If the tires check out OK, shocks or suspension bushings can give a wobbly/wiggly ride.
Tires first, that would be the most common.
Fishtail at speed? Check the trunk for fish. Check the trunk lid for “fish” or Phish stickers. Does the car quiver when you show it a worm? 
AskNott, you’re a riot!
Philster, about 180,000 miles.
Well I guess I’m off to go get the tires checked… I sure hope that’s the problem