So the situation has changed dramatically! I'm not an NF-expert even remotely, so I can't comment on what follows, but:

• Randall Holmes has now posted his proof of the consistency of NF: http://math.boisestate.edu/~holmes/holmes/nfisconsistentbytangledtypes.pdf. The proof is quite long, and to the best of my knowledge has not been fully vetted, but it was circulated privately for some time (as mentioned in the comments) so I am optimistic.

• While Holmes' proof was circulating, James Gabbay posted a proof of the consistency of NF: http://arxiv.org/abs/1406.4060. His proof is also quite long, but he has slides provide a nice (and funny!) summary of the argument: http://gabbay.org.uk/talks/20141022-leeds-2.pdf. EDIT: But see Andres' comment below.

• Holmes' (purported) proof is relative to much less than ZF, I believe to the theory TST which is roughly as strong as Zermelo set theory Z; Gabbay's (purported) proof is relative to ZF, but likely uses nothing beyond Z.

• The thread http://www.cs.nyu.edu/pipermail/fom/2014-July/thread.html#18026 at the mailing list FOM discusses both proofs, although not in detail.

So the situation has changed dramatically! I'm not an NF-expert even remotely, so I can't comment on what follows, but:

• Randall Holmes has now posted (EDIT: this is unintentionally misleading, see Thomas' answer below) his proof of the consistency of NF: http://math.boisestate.edu/~holmes/holmes/nfisconsistentbytangledtypes.pdf. The proof is quite long, and to the best of my knowledge has not been fully vetted, but it was circulated privately for some time (as mentioned in the comments) so I am optimistic.

• While Holmes' proof was circulating, James Gabbay posted a proof of the consistency of NF: http://arxiv.org/abs/1406.4060. His proof is also quite long, but he has slides provide a nice (and funny!) summary of the argument: http://gabbay.org.uk/talks/20141022-leeds-2.pdf. EDIT: But see Andres' comment below.

• Holmes' (purported) proof is relative to much less than ZF, I believe to the theory TST which is roughly as strong as Zermelo set theory Z; Gabbay's (purported) proof is relative to ZF, but likely uses nothing beyond Z.

• The thread http://www.cs.nyu.edu/pipermail/fom/2014-July/thread.html#18026 at the mailing list FOM discusses both proofs, although not in detail.

So the situation has changed dramatically! I'm not an NF-expert even remotely, so I can't comment on what follows, but:

• Randall Holmes has now posted his proof of the consistency of NF: http://math.boisestate.edu/~holmes/holmes/nfisconsistentbytangledtypes.pdf. The proof is quite long, and to the best of my knowledge has not been fully vetted, but it was circulated privately for some time (as mentioned in the comments) so I am optimistic.

• While Holmes' proof was circulating, James Gabbay posted a proof of the consistency of NF: http://arxiv.org/abs/1406.4060. His proof is also quite long, but he has slides provide a nice (and funny!) summary of the argument: http://gabbay.org.uk/talks/20141022-leeds-2.pdf.

• Holmes' (purported) proof is relative to much less than ZF, I believe to the theory TST which is roughly as strong as Zermelo set theory Z; Gabbay's (purported) proof is relative to ZF, but likely uses nothing beyond Z.

• The thread http://www.cs.nyu.edu/pipermail/fom/2014-July/thread.html#18026 at the mailing list FOM discusses both proofs, although not in detail.

So the situation has changed dramatically! I'm not an NF-expert even remotely, so I can't comment on what follows, but:

• Randall Holmes has now posted his proof of the consistency of NF: http://math.boisestate.edu/~holmes/holmes/nfisconsistentbytangledtypes.pdf. The proof is quite long, and to the best of my knowledge has not been fully vetted, but it was circulated privately for some time (as mentioned in the comments) so I am optimistic.

• While Holmes' proof was circulating, James Gabbay posted a proof of the consistency of NF: http://arxiv.org/abs/1406.4060. His proof is also quite long, but he has slides provide a nice (and funny!) summary of the argument: http://gabbay.org.uk/talks/20141022-leeds-2.pdf. EDIT: But see Andres' comment below.

• Holmes' (purported) proof is relative to much less than ZF, I believe to the theory TST which is roughly as strong as Zermelo set theory Z; Gabbay's (purported) proof is relative to ZF, but likely uses nothing beyond Z.

• The thread http://www.cs.nyu.edu/pipermail/fom/2014-July/thread.html#18026 at the mailing list FOM discusses both proofs, although not in detail.