For a reductive group defined over a p-adic field, the wavefront set is an invariant of an admissible representations which roughly speaking measures the direction of the singularities of the character near the identity. Studied first by Roger Howe in the 70s, the wavefront set has important connections to Arthur packets, and has been the subject of thorough investigation in the intervening years. One of main open lines of inquiry is to determine the relation between the wavefront set and the L-parameter of a representation. In this talk I will present new results answering this question for unipotent representation with real infinitesimal character. The results are joint with Dan Ciubotaru and Lucas Mason-Brown.