Lensfree on-chip imaging and sensing platforms provide compact and cost-effective designs for various telemedicine and lab-on-a-chip applications. In this work, we demonstrate computational solutions for some of the challenges associated with (i) the use of broadband, partially-coherent illumination sources for on-chip holographic imaging, and (ii) multicolor detection for lensfree fluorescent on-chip microscopy. Specifically, we introduce spectral demultiplexing approaches that aim to digitally narrow the spectral content of broadband illumination sources (such as wide-band light emitting diodes or even sunlight) to improve spatial resolution in holographic on-chip microscopy. We also demonstrate the application of such spectral demultiplexing approaches for wide-field imaging of multicolor fluorescent objects on a chip. These computational approaches can be used to replace e.g., thin-film interference filters, gratings or other optical components used for spectral multiplexing/demultiplexing, which can form a desirable solution for cost-effective and compact wide-field microscopy and sensing needs on a chip.