Duotrope.com is a great place to research fiction publications of interest to you. You may find many publications of which you were unaware. Duotrope's fiction home/search page has a database of about 2825 publications at present. You can search with various filters, such as genre, theme, length, media, pay scale and others.

I decided to search the database for various genres, with the length set to flash. The result is shown in the table below. Adding up the various genres may not be useful since many publications publish multiple genres. This doesn't guarantee that all publications found have ever or ever will publish flash fiction, but at least they are not officially opposed to it.


Flash Fiction Publications by Genre

All genres	1158
Mainstream	382
Experimental	267
Fantasy	176
Science Fiction	169
Horror	162
Magical Realism/Surrealism	123
Cross Genre/Slipstream	119
Mystery	57
Crime/Suspense	40
Action/Adventure	30
Erotica	23
Romance	16
Western	9