I'm basing this answer off an article: Kevin Lepot, Karim Benzerara, Gordon E. Brown Jr., Pascal Philippot (2008). 'Microbially influenced formation of 2.7 billion-year-old stromatolites'. Nature Geoscience Vol.1 No.2, pp.118–121. DOI: http://dx.doi.org/10.1038/ngeo107
Instruments that are desirable (and are actually used) to study stromatolites/freshwater microbialites:
- Raman microspectroscope
- Confocal laser scanning microscope
- Scanning transmission X-ray microscope
- Transmission electron microscope (TEM), including high-resolution TEM
- Near-edge X-ray absorption fine structure spectroscope
- The usual sets of fine analytical chemistry and microcutting tools
As you might suspect, this may require rather massive, power-hungry and bulky rovers. There is a solution and it's called sample return. Although the tools and instruments are constantly evolving and undergoing miniaturization, a skilled and attentive lab researcher will find out more back here on Earth than a rover on Mars.
Another consideration in favor of sample return is the verifiability of possibly sensational results - the sample will be curated and subdivided between universities and labs worldwide, applying diverse techniques and not relying on a single set of ultra-costly but possibly systematically biased hardware.