Have there been any experimentally-verified systems of microRNAs targeting a gene set (e.g., in cancer, perhaps)?
Yes. Below is a link to a review of ncRNA (non-coding RNAs) and their role in disease. There are many examples in this review in all sorts of diseases, one of which is miR-200, which is thought to play a role in some cancers.
There are also some tables in the paper that list the miRNA and the disease they are linked to. They also have a reference for each one, so you could read more about that particular miRNA and its function, including the gene it regulates.
Im not sure if that article is open to the public or not. If you cant access it you could always check out the wikipedia entry for miR-200