In their paper on typosquatting, dated January 2010, Moore and Edelman write "complainants invoked the UDRP arbitration procedure more than 45 000 times", and cite "Brand Owners Could Have Prevented $220 Million In Domain Name Recovery By Spending $1 Million", a Corporation Service Company press release dated August 24, 2009.
This press release actually refers to "UDRP cases covering more than 45,000 domains". As readers know, a UDRP complaint against a respondent can target one domain name or more.
The ICANN page on UDRP statistics if frozen since May 2004.
How can one find the exact number of UDRP decisions? The most comprehensive search tool seems to be UDRPsearch (though it indexes only decisions from WIPO, NAF, ADR.eu and the late eResolution, but not CPR or ADNDRC). A blank search there leads to 33,202 decisions. Nevertheless, these results include "UDRP-like" proceedings, such as cases decided under the .au Dispute Resolution Policy for example. The website does not offer a function by which one could refine the results.
If you have suggestions to get a better approximation of the total number of UDRP decisions, or leads to other databases, please let me know!