Hi there! Sorry for just getting to this now, I've actually had this answer tee'd up for awhile.

The Quarantia was the body with the ultimate authority over criminal justice in Venice. Originally instituted in the twelfth century as an upper legislative chamber with broad additional powers on things as disparate as final appeals for court cases as well as monetary policy, by the thirteenth century all non-judicial powers had been transferred to the increasingly powerful Senate. Although the Senate was originally an advisory body collecting all the major officials of the state and the most respected members of the Great Council, it had over time become Venice's true legislature (while the Great Council, by sixteenth century, was only really assembled to appoint members to the aforementioned Senate and the higher offices of the state, the latter which almost always also entitled the officeholder to sit in the senate anyway).

The Venetians made no distinction between the branches of the government, complicating our attempts to frame them in a way familiar to us. In the realm of justice, the Senate maintained some twenty or so fanti, a word meaning "footmen." Although, perhaps unsurprisingly, most would be retired soldiers, the fanti were surprisingly similar to a modern police force. The fanti would generally stay in the marketplace in the Rialto; although their mandate was to examine transactions and catch swindlers, they were in reality mostly preoccupied with stopping thieves and pickpockets.

Helping the fanti were the Capi di Sestriere, or “Neighborhood Chiefs.” The highest political position to which a non-aristocrat could aspire to, the Capi were volunteers (although retired soldiers were preferred, the important thing was that they were respected community members) who would organize patrols of other volunteers and investigate crimes in their neighborhood, called a "Sestriere." Along with the Capi, the Signori della Notte also conducted what we would call “police duties.” A post normally assigned to patricians, translatable as “Gentlemen of the Night,” they would conduct investigations and trials for more serious crimes, typically those that occurred at night. They had enormously broad powers of search and seizure, although abuse of their office could be appealed to the Quarantia, who would, if the wronged party was deemed in the right, award a monetary compensation.

As in many other early modern legal system, the Signori's investigations depended greatly on support from the wronged party. Additionally, in this era before mass media, tight social networks meant word of mouth transmitted rumors rapidly through communities. The signori had broad powers of interrogation and could even demand things like a list of all clients staying at an inn at a given time.

But we have a surprisingly small amount of textual evidence for cases like homicide in Venetian courts. Most legal scholars were more interested in writing prestigious analysis on the political and legal minutiae of concerns perceived as being of high order. The rote and mechanical courts used by ordinary people were of little interest, and indeed the judges who presided over them were normally selected from the lower and less prestigious branches of the aristocracy. We thus have a lot more commentary on things like trials of corrupt officials in the provinces or the details of individual procedures in the Great Council than we do full documentation of murder investigations.

However, we do have a multitude of "working" documents to work with in one field: everyday trade. The haughty venetian aristocrats and the legal scholars they consulted might not want to get their hands dirty in shops and marketplaces, but they nonetheless recognized the enormous importance of everyday trade and transactions. So the venetian ruling class meticulously regulated commerce within the city. The office called Magistrature of Justice regulated trade and commercial law, including things like price controls in addition to the means and method whereby transactions occurred. This institution was commonly called the "Justizia Vecchia," or "Old Justice," perhaps better translated as "Old Courts;" and was organized as multitude of specialized tribunals presiding over matters like the sale of meat and poultry or grain and bakeries. The "Justizia Nuova" presided exclusively over taverns, nightlife, and the wine trade; a lucrative business in a mercantile city filled with rambunctious itinerant sailors. A tier above these courts were the "Provveditori," instituted in the 1560s as a sort of final court of appeal and overseer evaluating the quality of justice dispensed (as a testament to its role as both an oversight and appellate body, its appointees could be specifically chosen by the Senate, or "promoted" existing judges, typically nearing the end of their careers at the lower courts). Although this whole system produced a copious amount of documentation, it is often only transmitted to us in fragments.

A more in-depth analysis of the Venetian legal system is:

The Justice of Venice: Authorities and Liberties in the Urban Economy, 1550-1700 by James E. Shaw (although it mostly has to do with economic law)

While a good Italian-language examination is:

La giustizia e la politica nella Repubblica di Venezia, in Id., Repubblica di Venezia e Stati italiani. Politica e giustizia dal secolo XVI al secolo XVIII by Gaetano Cozzi

And the most extensive primary source on the 16th century Venetian legal system are the diaries and published works of Marin Sanudo, a Venetian intellectual and legislator.