I think Aer Lingus tend to prefer integrated as opposed to modular when recruiting whitetails (I may indeed be wrong) but that's about it other than the list ManUtd1999 put together.
Otherwise, I fail to really see the benefits of integrated unless you're backed up by an airline with a conditional employment contract at the very least..
You complete all of your training with one provider but you can do that with modular too. I hear at CTC people fly about 5 hours a week on average, surely at least 10 hours/week is possible right? I recall seeing the RAF Fighter Pilot documentary from 1981 and they were averaging 10 hours a week in fast jets!
The integrated-only airline list is nowhere near as long as it was. CTC's current partners for non-BA Whitetails are Wizz, Turkish and Royal Brunei, a quick look at the T&E forums on here doesn't portray any of them in the best light. No UK bases (except possibly Wizz at Luton in the future).
Far less flexibility with integrated, if the industry slows, then at least with modular you can stretch your training out over a longer period, likewkse, if things pick up you can probably smash more than 5 hours/week in.
The integrated schools can get funny about results when it comes to recommendations time, one second series pass can mean you're at the back of the queue, ot so I hear. I know modular can be e same but at least you can transfer or possbily alter your training to follow a slightly different path.
Out of interest, does anyone know what happens if you fail on a mentored programme, do you get any of your money back? Did the loan guarantee offer any additional protection in the BA/Virgin programmes?