Can women chant at Mass?

 
The short answers are:

In the Extraordinary Form, yes (no surprise there), but they are not to form part of a choir of clerics (a "clerical choir") situated in the sanctuary (inside the altar rails).  So, the choir should be placed outside of the sanctuary (and, typically being composed of men and women and usually often do not wear any particular vestments or robes).  The Church's norms from early in the 20th century that suggested the answer was "no", were actually saying "yes, provided..." (as above). 

In the Ordinary Form, the same answers should apply.

Grab a look at Jeffrey Tucker's recent article getting to the bottom of this here