I don't understand your question. Are you asking about a Web browser or a file browser? Or some other type of browser? As far as I know, most homebrew is compatible with the DSTT, as is the Opera Web browser. Does that answer your question?
There are a handful of homebrew Web browsers for the NDS. Some of them rely on a Windows PC to do the rendering (for speed) and then send the rendered pages to your NDS. These require a client for your NDS and a server that runs on your PC.
A few others attempt to render Web pages on your NDS, but most only support a subset of HTML although there are some that support images. They therefore render pages imperfectly. DSOrganize, for example, has a Web browser that renders only text and small images.
There is also the commercial Opera browser, which offers the most complete HTML implementation. Opera works pretty well, but as expected it's slow. Most Web pages aren't designed for a tiny window, and therefore you have to do a lot of scrolling around to see larger pages. It works, but it's painful. Also, Opera requires extra memory provided by a slot-2 expansion card such as the Ewin or the 3-in-1.
As far as I know, all of the Web browsers will work with the DSTT.