Hello!

You can choose what to constraint, such as rotation or translation or both, and more in the transofrm constraint settings:
Transform Constraints - Spine User Guide

Here's an example that gets as a result to constraint only the x axis:
How to set up a child's bone to move only on that axis?

You can also get creative and use other constraints properties to get particular effects like a sort of distance limiter:
making eyes look at a direction within a distance limit?

If you have something in particular you wanted to achieve I'd love to help