Ha, I've been a bit over the top. It's a fan forum.
This would require a long post that nobody would want to read. I'll state it this way: look at the fan reaction that other teams are having just this season to ridiculous calls (Dallas, Detroit, etc). Now imagine you were a witness to that over the 15 years that Kraft has been in charge. It starts with the first and only PI penalty called on the Bills during a Hail Mary against the Patriots in, I think, 1996. Absurd.
But it's really about pattern recognition that only a lunatic fan would recognize. It's about sitting through a game and the refs are absolutely hosing the Bills, only to suddenly recognize more fouls against the suddenly undisciplined Patriots once the game is out of doubt. This evens out the end-game statistics so people on forums can say "Ah, dude, the penalties were 9 to 8 so all even" but Football games are a story, not an outcome, and I firmly believe those stories are affected.
I also believe the Spygate issue was much worse than advertised. Occam's razor is often cited here on this forum, what other reason would there be for burning the tapes in 4 days? What purpose? Why would they do that? It's because clearly the integrity of the game was at stake (super bowl outcomes), and while you'd say what owners would go along with it, you must be reminded that the owners' values are all related to each other, if the ship goes down, they all are worth less (i.e. the losers are just a portion of the masses).
People will cite media retractions and other hogwash, but I categorically deny that based on the tape destruction. If they had nothing to hide, there would be nothing hidden.
The populace is catching on though...there were actually published articles about the absurd calls against the Bills this past season after the Patriots game on ESPN. 15 yard penalty after a sack on 3rd down on Jerry Hughes for slapping Marcell Dareus' helmet in celebration. I consider that Referee opportunism...it was not even close to a penalty, and far worse celebrating occurs after every sack, but it was a chance to alter the game in the favor of the Patriots and the ref took it.
Your question was about specific issues related to cheating by the Patriots' players and coaches...obviously I don't have those specific things. One area where I would bet you bottom dollar they are doing things is cheating around the salary cap. I'll grant they are well-coached and managed, but these are pro athletes and they still want to get paid. My guess with would be that's an area of interest.