Singular measure with respect to translates

Question

Let $\mu$ be some Borel measure on $\mathbf{R}$ such that, for every $t \neq 0$, the push-forward $(\tau_t)_* \mu$ is singular with respect to $\mu$ (where $\tau_t(x)=x+t$). What can we say about $\mu$? An obvious example is Dirac measure at some point. Are there any other measures satysfying this property?