Further to my thoughts about the contradiction between "multiple attestation" and "embarrassment" in Historical Jesus Research, I'd like to throw out a related thought that I think compromises the value of the "multiple attestation" criterion. Sometimes single attestation might in fact be a far better indicator of historicity than multiple attestation.
Mark is often rightly seen as more primitive in its representation of Jesus in comparison with Matthew, Luke and John. There are moments where Mark provides a rough witness to an earlier stage of the tradition, where single attestation is a better indication of historicity than multiple attestation would be. Where Jesus heals the Deaf Mute with physical agents (Mark 7.31-37) and where he heals the Blind Man of Bethsaida with some apparent limit to his abilities (Mark 8.22-26), these pericopae are not included by Matthew and Luke, no doubt because of the uncongenial picture of Jesus they imply. Here, single attestation is a better indicator of primitivity than multiple attestation would be. To put it another way, is the Multiplication of the Loaves, attested six times in the Gospel tradition, the more likely to be historical than Jesus spitting in the mouth of the deaf mute, which is attested only once?
I am aware, of course, that multiple attestation, as it is usually configured, refers to multiple independent attestation, and so one is searching for material that occurs in allegedly independent sources like Mark, Q, Thomas and John. Notwithstanding my scepticism about the existence of Q and the independence of Thomas, the point still stands by analogy between the written Gospels and attestation in traditions, that uncongenial material will often have dropped out, so that some of the more unusual, primitive features will be at best singly attested where they are attested at all. In this context, multiple attestation is actually a weaker indication of historicity than is single attestation.