Get a second person to be in the driver's seat and move the steering wheel back and forth rapidly. You inspect the tie rod ends, and the joint where the tie rod meets the drag link. With the rapid back and forth you'll easily see if there's any play in those steering components.
If they look good, check ball joints visually with the wheels off. Sometimes there isn't enough play to notice and it holds together fine at lower speed, but when you get up there all hell breaks loose (this was the case on mine just the other day when I had the same problem as you).
After those two things you can measure the width between the tires at both the front most point and the rear most point to ensure your allignment is on. The distance between the tires at the front most point should be 1/8" less then at the rear most point.
Check air pressures in tires. Did you loose a bead recently and get dirt in there? Don't know how hard you wheel and if that may have happened at some point.
It happens at 60mph... Whats the most common cause? So many posts...hard to know what to do first...I don't see anything worn....New Tires and alignment needed? Steering damper? Thanks
If I'm not mistaken, 60mph occurrence is commonly a wheel balance issue. Whatever it is, it's NOT the steering stabilizer.
Follow the advice above and/or read more after having balance done an checking wheels for trueness. (they can be both balanced and out of round or un-true at the same time, which can make dx-ing DW difficult sometimes.)
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Get a second person to be in the driver's seat and move the steering wheel back and forth rapidly. You inspect the tie rod ends, and the joint where the tie rod meets the drag link. With the rapid back and forth you'll easily see if there's any play in those steering components...
Also check the track bar while doing this. Watch both ends for any movement.
Had the same problem years back, verify that your steering linkages are tight, swap out the steering stabilizer (highly recommend using a pitman arm puller to remove the old one) get a wheel alignment and the tires balanced. Ultimately changing the SS did the trick. It only happened to me between 55 and 60mph.