  A circulant graph is a simple graph whose adjacency matrix can be represented
in the form of a circulant matrix, while a nut graph is considered to be a
graph whose null space is spanned by a single full vector. In a previous study
by Damnjanovi\'c [arXiv:2212.03026, 2022], the complete set of all the pairs
$(n, d)$ for which there exists a $d$-regular circulant nut graph of order $n$
has been determined. Motivated by the said results, we put our focus on the
quartic circulant graphs and derive an explicit formula for computing their
nullities. Furthermore, we implement the aforementioned formula in order to
obtain a method for inspecting the singularity of a particular quartic
circulant graph and find the concise criteria to be used for testing whether
such a graph is a nut graph. Subsequently, we compute the minimum and maximum
nullity that a quartic circulant graph of a fixed order $n$ can attain, for
each viable order $n \ge 5$. Finally, we determine all the graphs attaining
these nullities and then provide a full characterization of all of their
corresponding extremal null spaces.
