Saturn's moon Titan is the only extraterrestrial body known to host stable lakes and a hydrological cycle. Titan's lakes predominantly contain liquid methane, ethane, and nitrogen, with methane evaporation driving its hydrological cycle. Molecular interactions between these three species lead to nonideal behavior that causes Titan's lakes to behave differently than Earth's lakes. Here, we numerically investigate how methane evaporation and nonideal interactions affect the physical properties, structure, dynamics, and evolution of shallow lakes on Titan. We find that, under certain temperature regimes, methane-rich mixtures are denser than relatively ethane-rich mixtures. This allows methane evaporation to stratify Titan's lakes into ethane-rich upper layers and methane-rich lower layers, separated by a strong compositional gradient. At temperatures above 86 K, lakes remain well mixed and unstratified. Between 84 and 86 K, lakes can stratify episodically. Below 84 K, lakes permanently stratify and develop very methane-depleted epilimnia. Despite small seasonal and diurnal deviations (<5 K) from typical surface temperatures, Titan's rain-filled ephemeral lakes and "phantom lakes" may nevertheless experience significantly larger temperature fluctuations, resulting in polymictic or even meromictic stratification, which may trigger ethane ice precipitation.