Consider:

 f = 0.5 (x^2 + y^3);
 ContourPlot[f, {x, -1, 1}, {y, -1, 1}, 
 MeshFunctions -> {Function[{x, y, z}, x], 
 Function[{x, y, z}, 3 y^2/2]}, Mesh -> {{0}, {0}}, 
 MeshStyle -> {Directive[Thick, Yellow], 
 Directive[Red, Thick, Dashed]}]

Which produces this image.

![enter image description here][1]

Why doesn't the Red, Thick, Dashed mesh show up?

**Example showing why it will not just be lines:**

 f[x_, y_] = x^2 - 4 x y/(y^2 + 1);
 ContourPlot[f[x, y], {x, -2, 2}, {y, -2, 2},
 MeshFunctions -> {Function[{x, y, z}, Evaluate@D[f[x, y], x]],
 Function[{x, y, z}, Evaluate@D[f[x, y], y]]},
 Mesh -> {{0}, {0}},
 MeshStyle -> {Directive[Thick, Dashed, Red], 
 Directive[Thick, Yellow]}]

Which produces:

![enter image description here][2]


 [1]: https://i.sstatic.net/WDvxJ.png
 [2]: https://i.sstatic.net/nTR4r.png