There are many accreditations used by outsourcing companies but there is 1 which often makes companies sweat and that is COPC. COPC is an independent organisation which assists in the training and development of call centre companies and provides an accreditation. In The United States and in parts of Asia where they service The United States, the use of COPC accreditation is fairly common. However, in The UK, it's rarely heard of. Outsourcing companies generally don't like COPC. They will typically only receive an accreditation specific to one or more clients when the client insists on it. It's expensive to implement and administer and is not as well known as other more common accreditations such as the ISO family of certifications. The accreditation also receives some negative publicity as part of its very expensive tie-up with Aegis which also saw one of the founders (Peter Bloom) eventually go and work for Aegis.
However, COPC is a lot more than ISO 9001. Instead of being simply a check-list of things to do, it is a process of improving performance and this is its key selling point and the reason why some companies have been keen to invest. COPC also offer "quality process outsourcing" otherwise known as QPO which in effect is outsourcing your quality department. This has advantages including more objective quality assessment by using a 3rd party but offers very little over and above what a company could do themselves for lower costs internally.
I never advise my clients to insist on COPC accreditation but I can see why it is useful as a differentiator in the vendor selection process.