I'm looking for statistics on the probability of a given opening being used in a game, and by what percent its use is growing or declining over some set timeframe, for example, over the previous year. How do I do this?
Chessgames.com is an excellent website to view statistics on opening moves and analyze games overall.
Any chess database (chessbase online, for example) can give you general statistics on how often a move is played in a given position, which you can use to get a feel for how popular an opening is. To see an opening's popularity over time, and other more sophisticated metrics, you could use a commercial product like ChessBase, which has options like 'Opening Report' which gives you graphs of frequency over time, top player games in the position, and the like.
The old surveys in the New In Chess Yearbooks used to have that for the opening being surveyed, but they've dropped that in recent years. Some old books by Elsvier used to continue the tradition of the ancient and long out of print Blue Book of Charts by printing frequency along with won/lost statistics of each move.
If you're willing to settle for recent years, Mark Crowther's fantastic This Week In Chess series would be useful. Every week he publishes for download a package of games played in international chess that week. Download them, load them in to SCID, and you're on your way, for an investment of no extra money.
For games older than that, if you wanted to keep the cost at free, you could try the University of Pittsburgh Archives. They have lots of games, just be aware there has been little or no QA work done on them, so there will be lots of duplicates, as well as fictional games.
Better QA work has been done for both the usual suspect (Chessbase) and for Chess Assistant where you can bring in 5+ million games for under $100US, but you'll have to pay for the editing. Even then, these games collections aren't perfect, just better. With the commercial databases you can walk the opening tree and see popularity of moves.
I'd caution you, though, popularity of moves doesn't yield good moves. If a given position occurs 30 times, then one side discovers a tremendous advantage, the advantageous move will only get played a couple of times, while the weaker moves may have been played a dozen times or more.