Avant Browser 10.1, build 40, works under Wine 0.9.10 with a few caveats. The 'Address:' bar jiggles (in both the View->Address Bar and 'Open a link' window). The 'Address:' bar has to be turned off (uncheck the View -> Address Bar checkbox) in order for the browser to start up fully. The up/down rail aren't properly painted, but you can navigate a 'cut-off' page with the up/down or page-up/page-down keys. But in general it works.
Things appear to be moving backwards with Avant Browser 11.6, build 6, on Wine 1.0-rc3. The browser installs, but running it is a bit of a downer since clicking on a link doesn't work nor does typing on a webpage.