  We say that a discrete set $X =\{x_n\}_{n\in\dN_0}$ on the half-line $$0=x_0
< x_1 <x_2 <x_3<... <x_n<... <+\infty$$ is sparse if the distances $\Delta x_n
= x_{n+1} -x_n$ between neighbouring points satisfy the condition $\frac{\Delta
x_{n}}{\Delta x_{n-1}} \rightarrow +\infty$. In this paper half-line
Schr\"odinger operators with point $\delta$- and $\delta^\prime$-interactions
on a sparse set are considered. Assuming that strengths of point interactions
tend to $\infty$ we give simple sufficient conditions for such Schr\"odinger
operators to have non-empty singular continuous spectrum and to have purely
singular continuous spectrum, which coincides with $\dR_+$.
