I found a published paper that looks dodgy. What to do?

A good way of posting comments, anonymously or not, with a good chance that other people will read them and react is Pubpeer. Although not all fields are covered, you can post your criticism on many articles. Sometimes authors react, sometimes publishers take action, sometimes nothing happens.

what leverage is usually in process for a retraction to happen?

A good list of the cases where Pubpeer comments lead to retraction can be found on the Retraction Watch website:

http://retractionwatch.com/?s=pubpeer

IMHO, Ignore it and move on

With the outbreak of many journal publishers and reviewers of diverse levels of experience to review them, you would find many such papers being published today. As a researcher it would be up to you to determine which paper seems verifiable and of proposed results are reproducible.