I have a curve shape converted to mesh and extruded, and now I want to bevel the perimeter. However, the bevel ends up messing up at certain places and I do not know why. Why one bunny ear and not the other? Why one point and not the other? I looked at a few different answers that said make sure your curves are clean, and I believe mine are. Thank you so much!
You have a messy topology that will prevent any good bevel.
Once you have converted to mesh, select all in Edit mode and press X > Limited Dissolve. In the Operator box choose a Max Angle value around 5°:
Create these edges at the bottom of the ears, clean a bit at the tip of the right ear:
Now the bevel works well, maybe it will need some correction at the tip of the ear, depending on the width: