Why don't the contrails show the curvature of the Earth?
Some contrails disperse and therefore end up being short to show any curvature.


(contrailscience.com)

Long ones don't stay put because of the wind. They wiggle.

It's for the same reason flying at 30,000 feet the Earth's curvature is not apparent. Earth is huge and it takes higher altitudes for the curvature to become noticeable to our eyes.



(californiacapitalairshow.com)

Flying higher at or above 60,000 feet or so (where the Concorde cruised) the Earth curvature starts to become more noticeable.

To notice it from the ground, you need a very clear day, not the slightest haze on the horizon, calm winds, and the plane to come from the right angle, as the illustration below shows:

(metabunk.org) 5-mile grid at 35,000 feet seen from the surface.
Note that the lines are squashed together the farther they are, so it'll take a very long contrail at a low angle above the horizon (called an altitude in astronomical terms) for that curvature to be easily seen.
RE comments:

The horizontal lines aren't as curved:

That's a property of one-point perspective, you see more depth than breadth.

The contrails are at least a few hundred miles long:

Not when viewed moving side-to-side. Jetliners cruise at around 550 MPH. The grid above shows what 5 miles are like. A 250-mile long contrail would take a plane 30 minutes to draw. You can time the next plane you see for proof on how long it takes to pass side-to-side in your field of vision. A lot less than 30 minutes. Example below is from a time lapse video:

Click image to open the GIF.
The GIF duration is 1.29 seconds and the video is shot at 5-second intervals at 24 fps. The contrail moved right to left in ~2.5 minutes, or roughly 25 miles (5 squares).