In The Wolverine, with his healing suppressed, there should be bleeding when Logan pops his claws.
Why does he bleed from bullet wounds, but not his hands?
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Per this answer:
Claws are extended through a silicon plug located in each knuckle cover by surgically-rearranged muscles. Silicon seals must be periodically replaced.
It's not a great answer, and there are some pretty clear differences between the claws on the trading card, and the claws as they appear in the movie, but its the best answer you're likely to find with some sort of canon authenticity.
If you're going for pure speculation, Paul D. Waite's answer is good. If the tissue covering the blades is thin, it would likely be fed blood through capillaries, and any bleeding would be minimal, whereas gunshots to the torso would quite possibly hit larger blood vessels.
Alternately, it is possible that, as part of his mutation, Logan has small opening in his skin that are normally closed off, but separate when the claws come out.
Idea: under normal conditions, he’ll have a bit of blood around a gunshot wound whilst the healing factor does its work. It doesn’t prevent him from ever bleeding, it just prevents him from bleeding for long.
However, in the movies, we never see any blood when he pops his claws, not even the first time he did it as a child (if I remember correctly from X-Men Origins: Wolverine). So, as a mutant, it may well be that his body doesn’t have any blood vessels where his claws come out (or at least not enough to result in visible blood).
The blades make small wounds (very thin) so I think these wounds would heal very quickly (a fraction of a second). Therefore, there would simply be no time for the wounds to bleed.
I remember one cartoon episode with a mutant child being able to suppress powers, where wolverine seems to endure pain upon retracting his claws. So at least, with his healing suppressed, retracting/popping his claws should hurt him (but then again, characters are frequently reimagined...)
Like a lot of things in comics, I believe the exact workings for Wolverine's claws have changed over the years, but I distinctly remember one storyline in which Rogue and Wolverine are in Genosha and have lost their powers. They are on a train, Wolverine pops his claws (or maybe just one) in order to cut the lock off the train door, and he starts bleeding. Rogue is surprised, and Wolverine says something along the lines of "My healing factor is gone, it normally closes the wounds made by my claws as soon as they're made". So at least in that storyline, he doesn't have silicone seals or anything, the claws break his skin and his healing factor normally stops the bleeding instantly.