This paper analyzes the diurnal evolution of temperature inside and in the vicinity of a small scale trench in the martian soil. Although the work presented was motivated by the artificial trench to be dug during the Mars Polar Lander (MPL) mission, the results are also applicable to natural structures. The models are calculated for conditions of region close to the south pole, matching those of the landing site for the MPL. Two trench geometries are considered. One has a rectangular cross section which could be achieved under the ideal condition of no material falling back during digging and the other has a trapezoidal cross section with the side walls inclined at 45°. Both trenches have flat bottoms and are assumed to be long enough to allow a 2D treatment of the heat conduction in the soil. In the case of a rectangular trench it is found that the water frost accumulation is expected on a significant part of the inner surfaces of the trench irrespective of the season. Some condensation of carbon dioxide could also proceed if the thermal diffusivity of the soil is close to the lowest available estimates. Frost formation in a trench with inclined walls is also possible depending on the assumed value of thermal diffusivity. The amount of frost should be large enough to allow the imaging of its accumulation with the Robotic Arm Camera of the Mars Volatile and Climate Surveyor experiment payload on the MPL.