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.
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:
