The middle-infrared (mid-IR) spectral range hosts the most intense roto-vibrational absorption lines of many important molecules. Particularly, mid-IR spectroscopy constitutes a unique tool for identifying and quantifying molecular species through their mid-IR spectral fingerprints. Exploiting efficient ways to access the mid-IR is of supreme importance for spectroscopic applications and environmental monitoring. Supercontinuum (SC) sources have been a common way to access this spectral region, using femtosecond fiber lasers and optical fibers as the broadening medium. However, emphasis has recently shifted towards compact SC sources based on integrated platforms. Among the different materials studied for the integration of SC, stoichiometric silicon nitride, $\mathrm{Si_3N_4}$, is one of the most promising. With a transparency window extending from the visible to mid-IR, CMOS compatibility, and a mature fabrication technology combined with flexibility in design, $\mathrm{Si_3N_4}$ waveguides represent an attractive solution for on-chip SC generation. The thesis's main aim is to develop a tunable mid-IR SC source based on $\mathrm{Si_3N_4}$ waveguides that can efficiently reach the spectral region between 3 - 4 $\mathrm{\mu m}$, enabling mid-IR spectroscopy experiments. Towards this goal, nonlinear SC dynamics based on anomalous and all-normal-dispersion (ANDi) pumping regimes are explored in different waveguide designs. In the first part of the thesis, the development of a tunable mid-IR source is presented. Starting from a tunable turn-key femtosecond 2 $\mathrm{\mu m}$ fiber laser located in the anomalous dispersion regime of cladded waveguides, mid-IR soliton-induced dispersive waves (DWs) are generated. A synergy of proper waveguide geometry and pump positioning enables favorable dispersion profiles towards the mid-IR while suppressing the generation into the visible. Moreover, by switching to a waveguide with different dimensions, the DW can be tuned between the entire 3 - 4 $\mathrm{\mu m}$ spectral region, reaching record efficiencies up to 35%. The milliwatt output powers of the mid-IR DWs guarantee their utilization in gas spectroscopy. In the second part, the application of the on-chip generated mid-IR source in direct absorption spectroscopy is demonstrated for the first time. The most efficient DW generated at 3.0 $\mathrm{\mu m}$ is used to detect acetylene in a proof-of-principle spectroscopic experiment. However, due to DWs' limited bandwidth, parallel gas detection is not possible. By leveraging a waveguide that generates a DW centered at 3.5 $\mathrm{\mu m}$ and fine-tuning the pump wavelength, a larger bandwidth of the DW source is achieved, enabling simultaneous detection of multiple gas-phase species. Using the broad mid-IR DW from a single $\mathrm{Si_3N_4}$ waveguide, detection of acetylene, methane, and ethane is successfully demonstrated, with hundreds of ppm detection limit. Finally, a polarization-sensitive SC source based on an un-cladded $\mathrm{Si_3N_4}$ waveguide is studied. The waveguide is engineered to yield ANDi at 2 $\mathrm{\mu m}$ for the TM mode and anomalous dispersion regime for TE mode excitation. A flat and highly coherent ANDi SC or an octave-spanning soliton fission-driven SC source can be selectively generated by simply switching between the pump's polarization states. As a result, the SC has the potential for versatile applications, from broadband on-chip sensing to pulse compression.