Your question is tagged Java, but it looks like you're asking about type characters, which are still supported in Visual Basic but not widely used these days (nor for a long time).
This is a bit subjective, but I think it's fair to say: never
One scenario where you might want to prefix a variable name with $
is when writing JavaScript code to distinguish jQuery objects.
Edit
Regarding starting a variable name with a $
, the Oracle Java tutorials tell us:
A variable's name can be any legal identifier — an unlimited-length sequence of Unicode letters and digits, beginning with a letter, the dollar sign "$
", or the underscore character "_
". The convention, however, is to always begin your variable names with a letter, not "$
" or "_
".