Statistics describing dental services and usage rates are collected as part of the previously cited Medical Expenditure Panel Survey (MEPS topics: Dental Visits/Use/Events and Expenditures). Unfortunately, the publications that summarize this data often do not include procedure-level details, but instead focus on dental expenditure on the whole as a percentage of health spending.

One exception I’ve found is a paper from 2007 comparing detailed statistics from 1996 and 2004. Here are relevant excerpts:

The total number of procedures increased from about 496 million in 1996 to about 572 million in 2004.

Distribution of dental procedures
	1996	2004
Diagnostic	40.2%	42.5%
Preventive	29.0%	30.4%
Restorative	8.2%	6.9%
Prosthetic	7.3%	6.4%
Oral surgery	4.1%	3.3%
Periodontic	0.9%	0.9%
Endodontic	2.1%	1.7%
Orthodontic	6.8%	6.9%
Other*	1.5%	1.0%

Diagnostic includes exams and x-rays.
Preventative includes cleanings, flouride, sealants and recall visits.
Restorative includes fillings and inlays.
Prosthetic procedures include crowns, bridges, and dentures.
Periodontic procedures include gums.
Endodontic procedures include root canals.
Orthodontic procedures include braces.
Other includes any other dental service not included above.

Src:
Richard J. Manski and Erwin Brown. 2007. “Dental Use, Expenses, Private Dental Coverage, and Changes, 1996 and 2004.” Agency for Healthcare Research and Quality, Rockville, MD.

Co-author Richard Manski (rmanski AT umaryland DOT edu) has written a couple of foreward-looking papers for RAND. I’ve emailed him to ask if he is aware of any ~10-year forecasts for dental braces usage.
EDIT: Manski is unaware of any forecasts.

Dental raw data from the on-going MEPS survey is available here: MEPSnet/HC Trend Query. Raw data for orthodontic visits in 2012 is here, for example: 2012 Full Year Person-Level File. DVGEN12 – # GENERAL DENTIST VISITS 12. Assistance in interpreting the data would be helpful.