Green card relinquishments since 2010, when FATCA was enacted:

2010: 19,545 
2011: 17,267
2012: 17,775 
2013: 11,185 (estimate)

These statistics would presumably not reflect the number of green cards relinquished by mailing of the card along with a letter of relinquishment to the nearest US consulate or to the United States Citizenship and Immigration Services (former INS) or “informally” abandoned green cards.

Even if the individual is not complying with the terms of maintaining the green card for purposes of the US immigration laws, continuing to hold the card still counts for US tax law purposes. Under the US tax rules, once “resident” status is acquired, it is deemed to continue unless it is rescinded or administratively or judicially determined to have been abandoned.
 A “rescission” of resident status generally occurs when there is a final administrative or judicial order of exclusion or deportation of the individual.   An administrative or judicial determination of “abandonment” of resident status may be initiated by the foreign person, the US Immigration authorities or a US consular officer. In most cases, the individual initiates the abandonment of the card, usually by filing INS Form I-407.  
The relevant Treasury Regulations are found in Section 301.7701(b)-1(b)(3)
 “If the alien initiates this determination, resident status is considered to be abandoned when the individual’s application for abandonment (INS Form I-407) or a letter stating the alien’s intent to abandon his or her resident status, with the Alien Registration Receipt Card (INS Form I-151 or Form I-551) enclosed, is filed with the INS or a consular officer. If INS replaces any of the form numbers referred to in this paragraph or § 301.7701(b)-2(f), refer to the comparable INS replacement form number. For purposes of this paragraph, an alien individual shall be considered to have filed a letter stating the intent to abandon resident status with the INS or a consular office if such letter is sent by certified mail, return receipt requested (or a foreign country’s equivalent thereof). A copy of the letter, along with proof that the letter was mailed and received, should be retained by the alien individual. If the INS or a consular officer initiates this determination, resident status will be considered to be abandoned upon the issuance of a final administrative order of abandonment. If an individual is granted an appeal to a federal court of competent jurisdiction, a final judicial order is required.”