The DVI format is older than Postscript and cannot describe Postscript. It does have an extension mechanism, namely DVI special codes, but the DVI specification gives them no semantics and so they depend on special nonstandard extensions in the DVI renderer to be useful.
The dvips program did attach significance to certain DVI specials: it allowed Postscript code to be encoded using them, which are now known as Postscript specials. There are other DVI extensions, such as the Hypertex specials used by the hyperref package or the TPIC specials used by eepic. These extensions have been encoded using text, and initial characters used to identify the type of special have ensured that the DVI specials provided by different extensions have not conflicted. In a way, this has been good luck: DVI specials have nothing like a namespace mechanism to ensure conflicts don't occur.
Ulrike's answer mentions the pstricks package, which uses these Postscript specials, as do some other packages, such as the DVI renderer for beamer. But to DVI renderers that do not recognise the dvips encoding of Postscript, the use of DVI specials will be meaningless and typically will be ignored.
Since Postscript is not "native" to DVI, it makes sense not to embed it: for one thing, this is more memory efficient since the Postscript does not need to be escaped. The standard for this is to use the psfile DVI special introduced by dvips.