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(I am surprised that a lot of people seem to be making very heavy weather of this. Perhaps I'm missing some subtlety in the question, but here goes.)

The classification of CDVRs with residue field any given perfect field k is discussed in Chapter 2 of Serre's Local Fields. In particular:

Theorem II.2: Let R be a CDVR with residue field k. Suppose R and k have the same characteristic and that k is perfect. Then R is isomorphic to k[[t]].

Theorem II.3: For every perfect field k of characteristic p, there exists a unique CDVR (up to unique isomorphism) which is absolutely unramified [i.e., p is a uniformizing element] and has k as its residue field: namely W(k), the Witt vector ring.

Theorem II.4: Let R be a CDVR of unequal characteristic with perfect residue field k. Let e be its absolute ramification index. Then there exists a unique homomorphism W(k) -> R commuting with reduction modulo the maximal ideal. This is injective, and R is a free W(k)-module of rank e.

Thus the CDVRs with residue field Z/p are: Z/p[[t]] and the valuation ring of a totally ramified extension of Z_p. In particular, the set of such isomorphism classes is countably infinite, and there no moduli in any sense known to me.

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(I am surprised that a lot of people seem to be making very heavy weather of this. Perhaps I'm missing some subtlety in the question, but here goes.)

The classification of CDVRs with residue field any given perfect field k is discussed in Chapter 2 of Serre's Local Fields. In particular:

Theorem II.2: Let R be a CDVR with residue field k. Suppose R and k have the same characteristic and that k is perfect. Then R is isomorphic to k[[t]].

Theorem II.3: For every perfect field k of characteristic p, there exists a unique CDVR (up to unique isomorphism) which is absolutely unramified [i.e., p is a uniformizing element] and has k as its residue field: namely W(k), the Witt vector ring.

Theorem II.4: Let R be a CDVR of unequal characteristic with perfect residue field k. Let e be its absolute ramification index. Then there exists a unique homomorphism W(k) -> R commuting with reduction modulo the maximal ideal. This is injective, and R is a free W(k)-module of rank e.

Thus the CDVRs with residue field Z/p are: Z/p[[t]] and the valuation ring of a totally ramified extension of Z_p. In particular, the set of such isomorphism classes is countably infinite, and there no moduli in any sense known to me.