"Hyperbolic-like directions" has been a central theme in geometric group theory. Two notions are usually used to quantify what is meant by "hyperbolic-like directions", the notion of a contracting geodesic and that of a Morse geodesic. Since the property that every geodesic ray in metric space X is contracting or Morse characterizes hyperbolic spaces, being a contracting/Morse geodesic is considered a hyperbolic-like property. In more general spaces, the Morse property is strictly weaker than the contracting property. However, if one adds an additional "local-to-global" condition on X, then Morse geodesics behave much like geodesics in hyperbolic spaces. Generalizing work of Cannon, I will first discuss a joint result with Eike proving that for any finitely generated group, the language of contracting geodesics with a fixed parameter is a regular language. I will then talk about recent work with Cordes, Russell and Spriano where we show that in local-to-global spaces, Morse geodesics also form a regular language, and we give a characterization of stable subgroups in terms of regular languages.