Actually the checkride at a 141 school (the "End of Course" stage check) is not reported to the FAA either unless you pass. Technically these cannot be failed, but rather are "incomplete". You get some more training, and come back and complete it later...there is no pink slip, and the FAA only sees the final paperwork after you pass. The school keeps records, but it is only required to for three years and they do not normally report those to an airline or other employer. The PRIA act has no requirement to obtain training records from your student days.

However...since colgan put the spotlight on unreported checkride failures, I would expect to see some rule changes where 141 schools will either start reporting EOC failures to the FAA or will have to keep records for much longer and report such failures to future employers.

Because of that, I would probably go ahead and report an EOC failure to avoid the appearance of deceit if the rules later allow (or more likely require) the employer to access that info.

As far as an intermediate stage check at a 141 school...tough call, but I would consider that no different from a 61 student whose instructor simply decides he needs another lesson on x-wind landings or whatever. I would probably not write it down on an app, but maybe mention it in the interview just to CYA. The interviewer probably won't care.