Someone in the US recently flew on a plane (opens new window) with a spirit level to prove that the earth is flat. Instead he demonstrated a misunderstanding of physics.
Planes have an artificial horizon (or attitude indicator), which allows pilots to keep the plane level. These devices use a gyroscope, which always point in the same direction. There is a flat earth argument that if pilots relied on these devices, they would keep gaining altitude as the device would keep them flying in a straight line, rather than following the curve of the earth.
Of course, this argument ignores the specifics of the design of artificial horizon devices on planes. In reality, these devices have the ability to slowly reset themselves using gravity as a reference - and this reference, gravity, follows the curve of the earth. Wherever you are on the planet, gravity always points towards the centre of the earth.
Planes also keep at a constant altitude using autopilot, which is fed by altitude measurements from pressure sensors, GPS, etc.
Also, air pressure becomes less the higher you go, giving planes less lift. I'm not sure, but this alone may be enough to keep a plane at a fixed altitude in equilibrium.
A spirit level relies on gravity. Our expectation, if we understand the physics, is that if you follow the curvature of the earth and fly a plane perpendicular to gravity at all times, a spirit level will always show that you are level.
However, one suggestion seems to be that in order to follow the curve of the earth, pilots need to dip the nose of the plane, and this is what the spirit level should be able to measure. However, that that dip has to compensate for 2m of drop every 5km travelled. For a triangle with two sides of 5,000m and one of 2m, that results in a dip angle of 0.0229° - not the kind of angle you can see from a spirit level. In the video, the random movement back and forth of the bubble shows natural variance (presumably due to turbulence, etc) way larger than the effect that the experiment is meant to measure.
The standard physics joke here talks about a spherical cow in a vacuum - the idea that to solve these problems, you first simplify them. Of course, this question is not simple and there are many variables that complicate issues. Needless to say, a builder's spirit measure on a plane is not going to prove anything, one way or another.