Starting from the premise that the CFM56 series engines are very very good all I can really add is that I agree with Rainboe, rely on the "pop-up" function that will get your attention!
Historically the earlier versions as fitted to the classic 737's were prone to flame outs in heavy rain, now fixed by a re-designed spinner boss and higher flight idle N1s with start switches/engine anti-ice ON. Fan blade failures on the early -C1 variants, fixed by a redesigned blade. Failure or seizure of the No.3 bearing, fixed on the engines on the -NG series. This was usually an instantaneous failure but according to a friend who had it happen to him, not nearly as dramatic as practised in the sim. If the engine chip detectors, and subsequent oil analysis, are done correctly the imminent bearing failure should be detected and the engine is changed before it happens. I have heard of the oil pressure dropping below limits in the descent at flight idle, the engine was shut down IIRC. I have had a problem with one engine "hunting" (ie not as dramatic as surging) I can't remember whether the problem was with the electronics (FMC, A/T computer, or PMC (it was a -300)), or whether the problem was more mechanical, ie fuel pump or fuel monitoring.
The variants as fitted to the -NGs have built on the quality of the earlier models. I have read a document by GE or SNECMA about engine failures when the airframe has been struck by lightning in the vicinity of the engines. The only case I know of was when an -NG was struck by lightning on about an 8nm final (so c2400' AAT). The engine failed and the crew carried out a OEI operative landing. Subsequent interrogation of the FDR and other "black boxes" showed the engine to have stalled, which was common to the other cases reported by GE/SNECMA, and was presumed to be caused by a disrupted airflow into the engine!
Not really in the same vein, and I presume that you are in the "desert" so probably not relevant to you, but my company has introduced a new procedure when taking off with the Engine Anti-Icing ON. Basically we do a standing T/O, setting 70% N1, confirming the engines are stable, before hitting TOGA. This is because we have had a number of instances of the MC> AMBER COWL AI OFF lights, and subsequent RTOs. The problem is apparently caused by either a faulty batch of valves and/or contamination in the system caused by operations into "dirty" strips. NB this procedure is not to be confused with the periodic 70% N1 ice shedding procedure!
Apologies for the thread drift but hope there is something useful for you in there!
