Partially ordered sets are closely related to directed graphs, and partially ordered spaces have been studied in general topology quite a bit since there is a strong connection between ordered sets and topologies. I realize that this may not be exactly what you had in mind since you may have been concerned mainly with paths on such spaces, but I will answer this question since ordered topological spaces are very interesting and very natural. Besides, one may also consider paths on ordered topological spaces as order preserving continuous maps from $[0,1]$ to your space $X$. Furthermore, I suppose that you can probably generalize this notion to a "locally ordered topological space" or a "locally preordered topological space" so that you can consider $\mathbb{R}^{2}\setminus\{0\}$ with a notion of going clockwise and counterclockwise. I will briefly outline some basic ordered topological spaces here.

If $X$ is a topological space, then the specialization ordering on $X$ is the preordering $\leq$ on $X$ such that $x\leq y$ if and only if $x\in\overline{\{y\}}$. Furthermore, the specialization ordering on $X$ is a partial ordering if and only if $X$ is a $T_{0}$-space. It follows from the definition that $X$ is a $T_{1}$-space if and only if the specialization ordering $\leq$ coincides with equality. Thus the specialization ordering is next to useless for spaces satisfying separation axioms from $T_{1}$ and above. However, the specialization ordering is very important for analyzing spaces that do not satisfy the $T_{1}$-separation axiom.

As another example of an ordered topological space, an Alexandroff space is a topological space where the arbitrary intersection of open sets is open. The category of Alexandroff spaces is isomorphic to the category of all preordered sets where the morphisms between preordered sets are the order preserving maps. If $X$ is an Alexandroff space, then the specialization ordering gives you a preordering on $X$. If $X$ is a preordered set, then the collection of all downwards closed sets are precisely the closed sets in an Alexandroff topology on $X$. These correspondences give you the equivalence between the category of Alexandroff spaces and the category of partially ordered sets.

Also, people have endowed partially ordered sets with other topologies such as the Lawson topology and Scott topology (see the book Continuous Lattices and Domains or the book the Compendium of Continuous Lattices) for more details. The Scott topology almost never satisfies higher separation axiom, but it turns out that the Lawson topology is compact and Hausdorff on a large class of ordered sets called continuous lattices.

Other ordered spaces that satisfy higher separation axioms are also useful.
For instance, one can use ordered compact Hausdorff spaces in order to study distributive lattices. We say that a topological space $X$ with a partial order $\leq$ is totally order disconnected if whenever $x\not\leq y$, then there is an upwards closed clopen set $U$ with $x\in U,y\not\in U$. A Priestley space is a compact totally order disconnected topological space. It turns out that the category of Priestley spaces is equivalent tot the category of all ordered topological spaces.

In my own personal research I have studied connections between least upper bounds in posets and topological spaces, and this research is quite similar to the Scott topology on a topological space. In my research, I have shown that a certain category of zero-dimensional ordered topological spaces is isomorphic to a category certain of pairs of the form $(X,\mathcal{A})$ where $\mathcal{A}$ is a collection of subsets of $X$ with least upper bounds.