Hide a weakness in ECC by choosing the prime or one of the curve coefficients
Bernstein and Lange says that there has been no progress for prime-field elliptic curves since about 1999, when the NIST curves were chosen. No large class of weak curves were known then, and no large class is known now.

Some small classes are known, (as Neves says) the curves with small embedding degree and the anomalous curves (order $n$ equals the prime $p$), but all of these were known back then (though not all of them by everyone in 1985, the MOV attack was a real attack, and the anomalous curves were an attempt to block the MOV attack, I think). So these classes are hard to hit at random, and are easily detected anyway.

The problem of embedding trapdoors in elliptic curves has been studied by Edlyn Teske, but she uses curves over extension fields, not prime fields. (Her techniques rely on it being hard to find isogenies between curves, but it is possible to find curves with isogenies. You choose one curve over an extension field that is secure, another that is not secure, and an isogeny between them. The isogeny is the trapdoor and allows you to move the DLOG instance from one curve to the other. The extension field attacks this relies on do not apply to NIST's extension field curves, as far as I know.)