We study the nonlinear stochastic time-fractional diffusion equations in the spatial domain $\mathbb{R}$, driven by multiplicative space-time white noise. The fractional index $β$ varies continuously from $0$ to $2$. The case $β=1$ (resp. $β=2$) corresponds to the stochastic heat (resp. wave) equation. The cases $β\in \:]0,1[\:$ and $β\in \:]1,2[\:$ are called {\it slow diffusion equations} and {\it fast diffusion equations}, respectively. Existence and uniqueness of random field solutions with measure-valued initial data, such as the Dirac delta measure, are established. Upper bounds on all $p$-th moments $(p\ge 2)$ are obtained, which are expressed using a kernel function $\mathcal{K}(t,x)$. The second moment is sharp. We obtain the Hölder continuity of the solution for the slow diffusion equations when the initial data is a bounded function. We prove the weak intermittency for both slow and fast diffusion equations. In this study, we introduce a special function, the {\it two-parameter Mainardi functions}, which are generalizations of the one-parameter Mainardi functions.