Static spherically symmetric uncoupled scalar space-times have no event horizon and a divergent Kretschmann singularity at the origin of the coordinates. The singularity is always present so that non-static solutions have been sought to see if the singularities can develop from an initially singular free space-time. In flat space-time the Klein-Gordon equation $\Box\ph=0$ has the non-static spherically symmetric solution $\ph=\si(v)/r$, where $\si(v)$ is a once differentiable function of the null coordinate $v$. In particular the function $\si(v)$ can be taken to be initially zero and then grow, thus producing a singularity in the scalar field. A similar situation occurs when the scalar field is coupled to gravity via Einstein's equations; the solution also develops a divergent Kretschmann invariant singularity, but it has no overall energy. To overcome this Bekenstein's theorems are applied to give two corresponding conformally coupled solutions. One of these has positive ADM mass and has the properties: i) it develops a Kretschmann invariant singularity, ii)it has no event horizon, iii)it has a well-defined source, iv)it has well-defined junction condition to Minkowski space-time, v)it is asymptotically flat with positive overall energy. This paper presents this solution and several other non-static scalar solutions. The properties of these solutions which are studied are limited to the following three: i)whether the solution can be joined to Minkowski space-time, ii)whether the solution is asymptotically flat, iii)and if so what the solutions' Bondi and ADM masses are.Comment: 25 pages, no diagrams or tables, LaTex2