Nontrivial 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 $n$.