  We characterize heat dissipation of supported molybdenum disulfide (MoS$_2$)
monolayers grown by chemical vapor deposition by means of ambient-condition
scanning thermal microscopy (SThM). We find that the thermal boundary
conductance of the MoS$_2$ monolayers in contact with 300 nm of SiO$_2$ is
around 4.6 $\pm$ 2 MW m$^{-2}$ K$^{-1}$. This value is in the low range of the
values determined for exfoliated flakes with other techniques such as Raman
thermometry, which span an order of magnitude (0.44-50 MW m$^{-2}$ K$^{-1}$),
and underlines the dispersion of measurements. The sensitivity to the in-plane
thermal conductivity of supported MoS$_2$ is very low, highlighting that the
thermal boundary conductance is the key driver of heat dissipation for the
MoS$_2$ monolayer when it is not suspended. In addition, this work also
demonstrates that SThM calibration using different thicknesses of SiO$_2$,
initially aimed at being used with bulk materials can be extended to 2D
materials.
