Rather surprisingly, the technique of convex integration has revealed that for some initial data, many models studied in the field of mathematical fluid mechanics allow for a multitude of solutions. This talk is concerned with the question how large the set of such wild initial data is in the context of the isentropic Euler equations. We show that wild initial data form a dense set. In contrast to existing results in the literature, in this talk "wild initial data" are data which give rise to infinitely many global-in-time weak solutions which are admissible in the sense that the local energy inequality holds.