5.0 Mustang fishtailing in a straight line

Only on snow without ABS. Not so much with ABS because of not having both working ABS and a working parking brake at the same time. I pull the parking brake to simulate a spin out and I start to go sideways. I lock the service brakes and then the car goes in the same direction while pointing off the road. Releasing the parking brakes frees the rear wheels and the back straightens out.

I know everything happens much faster on dry pavement.

I’m not familiar with the Ford traction control but some manufacturers us the brakes to control wheel spin so fixing your ABS may help resolve your problem.
As others have noted, having fresh “sticky” tires will also help.

On the other hand, a high power car with RWD is practically guaranteed to slip/chirp/squeal during rain or snow during anything remotely approaching hard acceleration.
The solution is simple, drive within your capabilities and don’t ask your car to make up the difference.

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My poor classic truck, now my daily driver, given its emissions era vintage, true enough, may only sport 130 HP. But remember who is driving. Even if it had 500 HP, any burnouts at all seems unlikely …lol …