At what point in time does a person's sense of self (or soul) arrive when using a transporter? Is there any time where a person would have no sense of self during transport?
In the Next Generation episode "Realm of Fear" we saw a transporter cycle from Reginald Barclay's point of view. Barclay was aware of himself and his surroundings throughout transport. So we can conclude that the sense of self is always present during transport.
However, in the episode "Relics" we learned that Scotty kept himself alive for over 75 years inside the transport buffer without sensing the passage of time. So while the sense of self is always present the time sense (the rate at which the brain state changes) can be altered during transport.
While @KyleJones's answer is the first thing to pop into my head, there's also a short bit of dialogue at the end of ENT 2x10, Vanishing Point, that helps here:
Trip: "You were sorta... trapped in the pattern buffer. But only for a few seconds."
Reed: "8.3 seconds, to be precise."
Hoshi: "Are you saying that I was just on the surface?"
Trip: "You insisted on going second?"
And in the next scene...
Hoshi: "So you're saying all of that happened in 8 seconds."
Phlox: "Actually it probably happened in the last, uh, 1 or 2 seconds, as your matter stream was coalescing. [turns to Archer] She seems fine."
So this would seem to imply that while stored in the pattern buffer (as Scotty was in Relics), your consciousness (and sense of self along with it) are inert, and you experience nothing.
(Aside, Hoshi's transition from real life -> pattern buffer -> dream -> real life was seamless from her perspective, which also explains why we didn't see this "inert" state while viewing the transport from Barclay's perspective in Realm of Fear)