Is there something similar to curves' tilt for skin modifier? I want to create shoelaces with skin modifier. I use Ctrl-A Y to flatten cross-section, and all looks good, but when I start to move vertices the angle of flattened part gets distorted and it does not confirm to the shoe surface. For curves I would use Tilt to solve this problem, but I do not know if something like this is possible for skin modifier as well.

Is there a 'tilt-like' way to control skin modifier? Am I better off to use curves in this situation?
From the very nature of the Skin modifier, this is not possible. Vertices do not have any rotational data, only location data. Because of this, it would be impossible to "tilt" the Skin modifier result, as it uses vertices to interpolate the resulting mesh.

Most of the examples I have seen the modeller is using curves. By using extrusion and bevel you can make great flat laces, and round ones are even easier. They also are completely editable at all times. 

I would strongly recommend using curves for this.

There is one example here that you can see. There is also a solution for one of the issues you might run into.