Wave is a code-based digital signature scheme. Its hardness relies on the unforgeability of signature and the indistinguishability of its public key, a parity check matrix of a ternary (U, U + V)-code. The best known attacks involve solving the Decoding Problem using the Information Set Decoding algorithm (ISD) to defeat these two problems. Our main contribution is the description of a quantum smoothed Wagner's algorithm within the ISD, which improves the forgery attack on Wave in the quantum model. We also recap the best known key and forgery attacks against Wave in the classical and quantum models. For each one, we explicitly express their time complexity in the function of Wave parameters and deduce the claimed security of Wave.