The answer is the $p$-regularisation of $(1^n)$: this is $(1^n)$ if $n < p$ and otherwise the partition $$ (r^a, (r-1)^{p-1-a}) $$ with $p-1$ parts, where $r$ and $a \in \{0,1,\ldots, p-1\}$ are defined uniquely by $$(r-1)(p-1) + a = n, $$ taking $a = 0$ if $p-1$ divides $n$.
For more on this see G. D. James, On the decomposition matrices of the symmetric groups II, J. Algebra 43 (1976), 45--54. The Mullineux Conjecture (now proved) determines the partition labelling a general $D^\lambda \otimes \mathrm{sgn}$ for $\lambda$ a $p$-regular partition. The shortest proof I know is in Example 24.5(iii) in James' Springer lecture notes on the symmetric groups.