This is a more technical question but it seems that there is some confusion in the literature on the choice of curves used to define the causal relations in time-oriented Lorentz manifolds: the infinitesimal causal relation is the choice of a forward light cone at every point, i.e. a time orientation. Now the (global) causal relation would take two points $p$ and $q$ and call $q$ in the future of $p$ if there is a causal curve from $p$ to $q$. Of course you have different futures depending on whether you take timelike or just causal curves, their interplay is of crucial importance in understanding the global geometry of Lorentz manifolds.

Now my question is what types of curves one actually uses:

One option is to stay with $C^\infty$-curves, simple but sometimes difficult in technical aspects as one would like to have arguments with broken geodesics etc.

The other option is to use piecewise $C^\infty$ curves, where the jumps of the velocities at break points stay in the same (forward) light cone. This should make the above disadvantage mostly disappear.

The option I have also seen in the literature (O'Neill...) is to take piecewise $C^1$-curves, again with the same condition at the break points. Now this is probably the most general version where one has still a meaning of what is "future-directed". However, many important(!) arguments in exploring the causal structure use variational formulas for arclength etc which can handle finitely many break points but require higher differentiability, say $C^2$ or even $C^3$.

So does the above choice matter? If so, what is the good convention?

EDIT: For convenience, here the more precise definitions: a tangent vector is called timelike if it is in the open interior of the lightcone and causal if it is inside the closed lightcone. It is called lightlike if it is in the boundary. Correspondingly, one has timelike, lightlike and causal curves, by using this for the tangent vector to the curve.

The set $J^+(p)$ is the causal future (we assume to have a time orientation) of the point $p$ which can be reached by causal curves, while $I^+(p)$ is the timelike future of the point $p$, i.e. those points which can be reached by timelike curves. While the condition "timelike" seems to be rather robust (tangent vectors inside the open cone) for causal this is more touchy. Nevertheless, for normally hyperbolic pdes on the Lorentz manifold, i.e. those with principal symbol given by the metric, it is the causal future $J^+$ which controls the propagation of singularities etc.

To give a flavour of known results (see e.g. O'Neill): for a causal piecewise $C^3$-curve $\gamma$ which can not be reparametrized into a lightlike geodesic one finds a variation $\gamma_s$ such that for all $s > 0$ the curve $\gamma_s$ is timelike and has the same starting and end point. This result is important for the transitivity of the causal relations. But the proof uses very much $C^3$.