If a painting is in a book or a magazine, the artist can be identified by referencing a table of contents or contributor's page. If the painting is on a canvas and the signature on the front is a monogram or illegible, the back of the canvas may have information.
If the full name of the artist is not on the painting or print, there are several resources that keep a database of artist's signatures and monographs. Websites, such as ArtistsSignatures.com, offer reverse look up searches to help identify illegible or unknown signatures. As of 2015, there are over 100,000 signature examples there that can be referenced for comparison.
Books, such as “The Encyclopedia of Artists' Signatures, Symbols & Monograms” by H.H. Caplan and Bob Creps, have several thousand artists' signatures and can be found in the reference section of any major library or art museum. If the painting is likely done by a local artist, visit local galleries and talk to curators who may have information about the artist or can recommend a local art historian. If none of these methods yields results, try searching for similar paintings on eBay or Amazon.com by using keywords that highlight specific details about the painting. For example, use the type of painting, the subject matter and the medium, such as landscape, sunflowers, oil. This may provide results from the same artist or even a replica print with the artist's name in the description.