Larry, I can address this, as I deal with the 1.4T in every car it's in (500 Abarth, Dart, 500L, MiTo, Giulietta, and 500 Turbo).
The basic engine and major components are basically bulletproof. The only components I have seen that seems to have an issue are the coil packs, and then only in the 500 Abarth and 500 Turbo's tighter engine bays. There have been a few coil pack failures in some other chassis, and I know of 1 in a 124, but it's pretty rare, and always covered by warranty.
The vast majority of problems have been owner induced, and I'll give some specifics. The Fiat engine uses some technology that's quite different from what was used in older Dodge Turbo cars. On the Fiat side, one vendor in particular had a lot of experience with another European car and stubbornly felt that the Fiat should be set up the same way. They just didn't understand the Fiat engine and they led a lot of people astray and caused many problems.
Here are two perfect examples. A lot of people screwed up the wastegate adjustment. This was because they didn't understand that the engine boost curve is designed to drop off at the high end. In order to "fix" the drop off they would tighten the wastegate adjustment. Then it wouldn't work so they would tighten it up more. Eventually, the engine wouldn't run correctly and they wouldn't be able to set it up correctly again. To make matters worse, the dealers often don't know how to adjust it either, and will replace the whole turbo to fix this! I have made a video to show people how to adjust it, so if you get a car that's been screwed up it can be fixed easily. However the key is just don't screw it up in the first place. These techniques worked with Neon SRT-4s but not on the Fiat.
The second example relates to the car's EVAP system. A small number of very early 2012 Abarth were sent out with a small plumbing issue with the system which was corrected by the dealers. (this physically can not happen on a 124) However a very large number of people hacked up the evap systems causing endless check engine lights. The EVAP system works perfectly, don't screw with it, and it won't be a problem. This specific system could be a major discussion by itself, I have a tech article on it on the eurocompulsion site if you are curious.
The bottom line is that the 1.4T is very reliable. I sponsor 6 race cars with this engine, all with over 200hp, some with over 250, none have suffered engine failures. However, it's NOT a Mazda engine, it's not a Neon, and it's certainly not a Mini. If you don't understand the engine and try and treat it like it's one of these other cars, it's going to cause problems which is what happened in the Dart community and early in the Fiat community.