Question: If $L$ is not regular and $ L \subset L\_1$, does it follow that $L\_1$ is not regular also? Can you please provide an explanation? Thanks in advance.

Answer: 
[@Vladislav's answer](https://cs.stackexchange.com/a/123316/4416) is probably more interesting, but observe that *every* language over an alphabet $\Sigma$ is a subset of $\Sigma^\*$, which is certainly a regular language.
No. Let $L$ be the language of balanced bracket sequences, and $L\_1$ be the language of arbitrary bracket sequences. Then $L \subset L\_1$, $L$ is not regular (you can prove it using [pumping lemma](https://en.wikipedia.org/wiki/Pumping_lemma_for_regular_languages)), but $L\_1$ is clearly regular.