You can pick up a copy of Pariscope at a new kiosk. It comes out weekly (Wednesday, I think) and gives a list of all concerts, etc for the following week.

Here's the Notre Dame schedule. I don't see a price so I'm guessing the musical events are free.

http://www.musique-sacree-notredamedeparis.fr/spip.php?article252

St Merri church has concerts on weekends.

St Eustache has organ concerts on Sunday afternoon.

I agree with what Nini said.

Just get an entertainment guide at a newstand, Pariscope or l'Officiel des Spectacles. They may not be on display, you can ask (they put them behind the counter often as I think people steal them). They are only around .50 euro.

They are in French, but you should be able to understand the listings easily enough (musique classique, concerts, etc.). Generally, there are a few each week that are free recitals in churches, usually, sometimes a few other places. Most tend to have some fee, even if not much (like 10 euro).

You will never find free opera or ballet. The production costs are very high which is why tickets cost so much.

You can also check ahead of time of www.whatsonwhen.com.

I can add some sites to look at although not necessarily free shows.
This is a useful list of performances by month at Saint-Julien-le-Pauvre in the 5th. http://www.concertinparis.com/US/USindex.htm
Here is a long (long!) list of links to museums and other locations. It's from 2009 but should not be too far out of date. http://www.parisnotes.com/museums/parismuseums.html
Radio France sponsors live music; here is the site, although in French http://sites.radiofrance.fr/chaines/formations/concerts/saison/calendrier.php

Pariscope (0.40€ at any newsstand -- comes out on Wednesday) lists all concerts in the city and tells the prices or non-prices of the events.

There are free concerts at the Madeleine church.

http://www.concerts-lamadeleine.com/