There are many sources for Responsorial Psalm music. Some are free and some really good ones may be purchased. In my desire to learn more about psalm chanting and singing, especially for the Responsorial Psalms in mass, I also searched the internet for free resources.
Chanting Psalms Workshop
There are actually several ways to chant or sing the psalms. One quick search turns up Gregorian chants. These manner of chanting sounds so very solemn and evokes an ambiance of chanting bowed monks in a monastery. Personally, they may not be suitable for the responsorial psalms in regular Sunday masses.
However, a quick google search for "Psalm Chanting Workshop" will yield youtube videos of how a simple style can still be effective and appropriate for Responsorial Psalms.
These videos explain in a clear manner how easy it is to chant psalms. Listed in their pdf file (seen in the video also) are the various psalm tones that can be used to match the mood of the psalm.
This particular free resource on the internet helped jumpstart my desire to becoming a serious psalmist. I still like the style which is so easy to adapt.
Free Responsorial Psalm Music Videos
In a previous article, I mentioned the Respond and Acclaim series that provide sheet music and mp3 sound files. The music covers the Responsorial Psalms and Gospel Acclamantions for the year.
Google search "Respond and Acclaim" on youtube and you'll find plenty of videos where different singers and musicians give their own renditions of this popular music.
I find the music of Respond and Acclaim to be very contemporary and melodic. Some are easy-listening music and don't sound like psalms at all. You may even be lucky to catch advanced videos of future Sunday responsorial psalms that were uploaded my these musicians.
Free Responsorial Psalm Music Sheet and MP3's
If you can dig deeper into the internet, you would find sheet music and sound files from other websites. Many church and parish websites actually store deep in their folders or directories tons of these sheet music in pdf's and sound files in mp3's.
You just have to know the proper internet google search tools to use. Yes, even the popular ones are available free somewhere.
Free Downloadable Sunday Missalettes on Internet
Firstly, the parish provides copies of the Sunday missalette for the parish workers for free. I suppose most parishes do that. Unfortunately, the copies are generally handed out on the day of the service itself. I doubt if they give them out (or even receive them) weeks in advance.
In addition to the parish's hardcopies of the missalettes, I've found a site on the internet that keeps advanced copies. I use that to get advanced copies for printing the responsorial psalms from the missalettes.