The glitch is due to numerical instability around a stationary point of the function. In Mathematica it can be eliminated, for example, by increasing the PlotPoints:
ContourPlot[x^y/y^x, {x, 2.6, 2.8}, {y, 2.6, 2.8},
PlotPoints -> 50]
I don't know what to do in Wolfram Alpha.
The 3D plot avoids the critical contour entirely.
