If $\mathbf{L^*}$ is regular, then $\mathbf{L}$ is not necessarily regular.
Let's take one possible example.
Let $\mathbf{\Sigma = \{a\}}$ and consider the language $\mathbf{L = \{ a^{2^n} \mid n \in N \} }$.
This language is not regular
and the language $\mathbf{L^*}$ is the language $\mathbf{a^*}$, which is regular. To see this, notice that since $\mathbf{L}$ contains the string a, the language $\mathbf{L^*}$ contains all strings of the form $\mathbf{a^n}$ for any natural number $\mathbf{n}$.
Hence, Option(2) Need not Regular is the correct choice.