Nontrivial examples of sparse languages A language $L$ is said to be sparse if there exist a polynomial $p$ such that $|L \cap \{0,1\}^n| \le p(n)$. 

One trivial example is suppose language is over single alphabet $1$ then $$L = \{1^n | n \in Z^+\}$$

Question : What are the other non-trivial  examples of sparse languages?
I don't think that there exists a reasonable characterization of all sparse languages (for instance, it's trivial to define sparse languages that sit arbitrarily high on the arithmetical hierarchy).

However, since $\{0,1\}^*$ has exponential growth, you need to be able to derive some kind of combinatorial property from the language definition. For example, let $k$ be a constant and consider the following language:

$\qquad L = \{x \mid x \text{ contains $k$ ones}\}$

$L$ is sparse, the key observation being that $\binom{n}{k}$ for a fixed $k$ is a polynomial in $n$.