Recently emerged layered transition metal dichalcogenides have attracted great interest due to their intriguing fundamental physical properties and potential applications in optoelectronics. Using scattering-type scanning near-field optical microscope (s-SNOM) and theoretical modeling, we study propagating surface waves in the visible spectral range that are excited at sharp edges of layered transition metal dichalcogenides (TMDC) such as molybdenum disulfide and tungsten diselenide. These surface waves form fringes in s-SNOM measurements. By measuring how the fringes change when the sample is rotated with respect to the incident beam, we obtain evidence that exfoliated MoS2 on a silicon substrate supports two types of Zenneck surface waves that are predicted to exist in materials with large real and imaginary parts of the permittivity. We have compared MoS2 interference fringes with those formed on layered insulator such as hexagonal boron nitride where only leaky modes are possible due to its small permittivity. Interpretation of experimental data is supported by theoretical models. Our results could pave the way to the investigation of surface waves on TMDCs and other van der Waals materials and their novel photonics applications.