I've written a couple of posts about some arguments against the existence of Q. One was the inclusion of Joseph in a non-Q pericope in Luke
. It seems as though Luke is following Matthew at this point (Lk 4.14-22 // Matt 13.53-55) and not Mark (6.1-3). There would be no reason for Luke to change Mark's “isn't this the carpenter” to “isn't this Joseph's son” without the intermediary — i.e. Matthew's — “isn't this the carpenter's son?”.
Another was the removal of the character Bartimaeus in Luke
. If Luke had been following Mark at Mk 10.46, then he should have included Bart's name as well. But he makes Bart's character anonymous just like Matthew does (Matt 20.30 // Luke 18.35). We know why Matt makes him anonymous, but why would Luke do it?
Another argument against Q is brought up by Mark Goodacre in his book The Case Against Q. We know that Matt used Mark due to editorial fatigue: one example is that Matt corrects Mark's use of “king” for Herod; the actual title was tetrarch (Mk 6.14 vs Mt 14.1). But due to editorial fatigue Matt lapses back to following Mark in the same pericope (Mk 6.26 // Mt 14.9) and “reverts” back to calling Herod a king.
The same type of editorial fatigue can also be seen in Luke's following of Matt. Luke usually has a 10-1 ratio in his parables/miracle stories (ten lepers, one thankful [Lk 17.11-19]; ten coins, one lost [15.8-10]). In the parable of the ten minas, Luke starts out with ten (Matt has three; cf Mt 25.14-30) but only recounts three servants by the parable's end, just like in Matt. And Luke seems to follow Matt's conclusion when he recounts giving the last mina to “the one who had ten”. In Matt it makes sense that there was one who had ten since he was given five at the beginning and doubled it. But in Luke, the one who had ten actually ends up with eleven (he starts out with one and adds ten more). So either Luke forgot how to add, or he is following Matt.
One of the big points that Goodacre makes against Q in the last chapter of his book is that it is a hypothetical document with no other exemplar. While we do have another list of sayings sort of document like the gospel of Thomas, Q actually has a narrative structure unlike Thomas. So again, Q is a hypothetical document that has no equal in any other early Christian literature. A list of sayings, but with a narrative structure: it begins with preaching in the wilderness by John (Q 3.2-7), to Jesus' implied baptism (Q 3.16-22), then being tempted by Satan (Q 4.1-13). Later on in Q, Jesus is approached by messengers from JtB (Q 7.18-35) which follows the same narrative structure as the gospels. It then refers back to JtB's prediction of the coming one mentioned earlier in Q.
Thomas has no logical progression like that and doesn't follow any narrative gospel that we know about. It looks to me, according to Goodacre's argument, that Q is also a narrative gospel (if indeed Q existed) that we just have fragments of and not a bare list of sayings like Thomas.
For me, this puts the prior for the existence of Q at a low probability. I would like to do some Bayesian analysis to see whether Q exists or not, but I'll probably save that for later as it would probably get really involved.