According to the book I am studying (Royden & Fitzpatrick):
We can infer from Zorn's Lemma that every linear space possesses a Hamel basis.
A Hamel Basis is defined as a subset $\mathcal{B} $ of a linear space $X$ provided each vector in $X$ is expressible as a unique $\bf{finite}$ linear combination of vectors in $\mathcal{B}$.
When I first read about the concept of Hamel bases (compared to other sets referred to as basis), I recognized that bases of finite spaces are Hamel basis, and that, for example, $\{x^k\}_{k=0}^\infty$ is a Hamel basis of $\mathbb{R}[x]$, the infinite-dimensional space of polynomials with real coefficients.
As comments have pointed out, there might not be a Hamel basis that is explicit to state for other common infinite-dimensional linear spaces ($\ell^p$ or $L^p$) - their Hamel bases would need to be uncountable.
However, even if we cannot explicitly characterize the whole basis, could we choose one that includes a specific (countable) set of vectors as a subset, given they are 'independent'? (For example, include $ \{x^k\}_{k=0}^\infty$ in the Hamel basis for $L^p([a,b])$). If so, what are the applications of this fact?
Yes, we can.
None.