Make a scan or take a photo of the artist's signature, if you don't already have a copy of it, to compare with signature examples in a directory or database.
Search through databases which offer samples of artist signatures. Some useful online resources are Artists' Signatures and the Online Art Research Center.
Create a free account on Artists' Signatures to search through a database of thousands of signatures by the artist's name or country.
Perform a reverse lookup of signatures if the name in the signature is illegible, a monogram, in a foreign script such as Cyrillic, or otherwise difficult to read. In addition to viewing an artist's name and signature, Artists' Signatures offers a complete profile of artists, including their date of birth.
Access the Online Art Research Center to look through a much smaller collection of signatures and fewer features without registering. When you access the site, you can click on an artist's name to view an image of their signature. This site also includes birth and death date information for each artist.
Browse through a directory of artists' signatures and monograms. Some good resources to use for 19th century artists are Artists as illustrators: an international directory with signatures and monograms, 1800 to the present; American artists: signatures and monograms, 1800-1989; Artists' monograms and indiscernible signatures: an international directory, 1800-1991 (all by John Castagno) and Die Monogrammisten by Georg Kaspar Nagler.
Contact an art appraiser or curator and send them an image of the signature if you still can't find a match.