This is an example of a question that we won't be able to answer until the CourtListener archive is complete. Using the data that is available, one could run queries there for particular statutes by citation or name.
For example, if interested in the Convention Against Torture, do a query at CourtListener for that with surrounding quotes, or if interested in cases invoking "42 U.S.C. § 1983" query that.
But cases aren't always "filed under" a particular statute. When filing a federal lawsuit one is required to fill out a cover sheet and choose a "Nature of Suit" (NOS) code that categorizes the case somewhat, but many lawsuits involve more than one issue, so a case alleging both copyright and trademark infringement might only be listed under the copyright NOS code and so even if you could search across all the NOS codes you wouldn't get exactly what you want. A separate problem illustrated by this example is that Copyright law is vast. Cases dealing with 17 U.S.C. § 107 (fair use) are often totally different from cases dealing with 17 U.S.C. § 109 (the first sale doctrine).
So, in some ways your question is not exactly framed in an answerable form. What could be done (when CourtListener's archive is complete) is we could create a giant list of every opinion that mentions "17 U.S.C. § 107" and every opinion that mentions "17 U.S.C. § 108", "17 U.S.C. § 109", etc. through the entire United States code. That would be really interesting. What is the most litigated part of the code? What statute is cited the most? etc.
The best collection of federal district court opinions that you could download/scrape would be the RECAP collection from the good people at CITP, hosted by the Internet Archive here: https://archive.org/details/usfederalcourts
However, that collection includes any kind of document that might appear on a federal docket, including complaints, briefs, declarations, etc. and so it would take some additional effort to sort out the opinions from the rest. One day CourtListener will ingest that collection and make this easier for people trying to tackle questions like yours.