Good day all,
Lightning near the hurricane core is also not very common because of the structure of the hurricane's core as well. In addition to being a warm-core system, one must also consider that the winds in the eyewall of a hurricane also decrease with height, until about half of what they are at 850 MB at 250 MB.
Any T-Storm cell in the eyewall will be highly sheared by this difference, allowing charge buildup on the ground nearly impossible (literally blows the field away) and leaving the only possibility of CC (cloud-cloud) lightning at high altitudes (near anvil level) in the eyewall.
The outer bands tend to behave more like "normal" thunderstorms / squall lines, and generate far more lightning than the eyewall.
Now, with all this said, eyewall lightning does exist. I have seen FREQUENT lightning in the eyewalls of intensifying hurricanes, mainly those at category-1 strength and tropical storms - Again, the less tilting of the updrafts in the core (due to the lighter flow) accounts for this. In hurricane Katrina, as it crossed S Florida as a cat-1 storm, I found frequent lightning in the eyewall, and there actually were a few CG's!
In hurricane Charley in Punta Gorda in Aug 2004 (strong category 4), I saw some occasional lightning in the worst part of the eyewall, just before entering the small "pinhole" 3-mile wide eye! During the second half, I saw a few flashes again in the eyewall - While hanging on and / or praying, ofcourse. Like RH said, hearing thunder is out of the question over the aircraft-velocity winds.
In Charley, the lightning detection revealed lots of lightning in the outer bands, then not much until you got near the eyewall, where a few "sporadic" blips once again appeared around the tight and small eyewall, especially in the convective "chimney" to the right-side of the eye / eyewall. The flashes I saw were NOT power flashes, there was clearly a flicker (return stroke), but I could not confirm if they were CC / CG (I assume CC) because of the poor "white-out" visibility.
I have seen eyewall lightning in hurricanes Frances and Jeanne in 2004 as well, and it was ruled-out as power flashes as it was seen east over the ocean. This lightning was "sheet" lightning, with a highly attenuated green/turquoise color ... Most likely due to the fact it was CC and very high up (looked like an anvil crawler type hidden by a stratus layer from plain view) with so many cloud layers between myself and the lightning bolt.
Some hurricanes, abeit intense, had no lightning. Wilma in 2005 had no lightning at all, not even in the feeder bands, let alone the eyewall. I have heard one report from Ivan back in 2004 that the hurricane hunters expoerienced "frequent lightning and moderate turbulence experienced in eyewall at 10,000 feet" ... I am wondering how much study has been done on this.