Is the air in a jet engine supercritical? Nitrogen has a critical point of 126.2 K and 3.4 MPa (34 bar). Air contains 75 % of nitrogen


  For example, nitrogen has a critical point of 126.2 K (−147 °C) and 3.4 MPa (34 bar).


source : https://en.wikipedia.org/wiki/Supercritical_fluid



In a jet engine pressures often exceed 34 bar ( as seen on Wikipedia, where for example the GE9X is shown to reach 60 times ambient pressure ), the critical temperature of nitrogen is exceeded at room temperature. So is air ( fluid ) in the high pressure regions of a jet engine supercritical ?
Maybe at takeoff, definitely not in cruise

You have to consider that atmospheric pressure drops with altitude. At FL360 you have ~0.226 bars. Even multiplying with the GE9X's pressure ratio of 60, you get ~13.56 bars, well below the 34 bars needed.  

Takeoff might be a different situation, since the engines are operating at full power with ambient pressure ~1 bar, this could lead the high pressure compressor stages to operate in supercritical conditions. Note though that the combustion that immediately follows changes significantly the chemical composition, and thus the critical point might be altered as a consequence.

As engines are optimized for cruise conditions, it is highly unlikely that supercritical conditions have extensive design consequences, unless they would render takeoff impractical.