At the moment our tests are limited to Hubble's law, measurements of isotropy, some indirect measurements and a bit of philosophy.
We observe that galaxies appear to move away from us, in an isotropic fashion, at a rate that is proportional to their distance from us.
Whilst one could argue that we are at (or near) the centre of this very uniform expansion it begs the question as to why Hubble's law should exist and why the universe appears isotropic to us, but wouldn't from a different position in the universe. The simplest explanation is that General Relativity applies (as we observe in a number of other cases) and we live in an expanding universe - this then means we do not need to occupy some privileged position in the universe (an erroneous assumption that has proved wrong every other time it has been made).
In such a universe, the redshift of distant galaxies is not caused by relative motion, but by the expansion of space. At high redshifts, these phenomena become distinct in that the relationship between "velocity" and redshift is different, for instance allowing "faster than light" (apparent) speeds.
So basically at present, expansion fits the facts (far) better and more simply than any of the alternatives.
A further piece of indirect evidence comes from a careful analysis of the physical conditions of gas at high redshifts, illuminated by background quasars and subtle alterations to the cosmic microwave background (CMB) spectrum, caused by the Sunyaev-Zel'dovich effect, towards galaxy clusters at low redshifts. Both of these methods give the temperature of the CMB at those locations.
In the expanding universe model, the temperature should increase as $1+z$, where $z$ is the redshift. If one instead has a non-expanding universe, and explain the CMB as due to some expanding shell of material, then the average temperature wouldn't change for distant galaxies unless the shell gas has been uniformly cooling by an amount that just happens to agree with redshift of that galaxy.
Avgoustidis et al. (2015) review the evidence for the temperature evolution of the CMB and conclude that it agrees with an adiabatic expansion to better than 1%.
Direct evidence for the expansion is on the horizon though. In an expanding universe, the speed at which galaxies move away from us can change slowly with time (and with distance) by of order 10 cm/s per year, despite their being no force on them. This is known as the redshift drift. There are plans to measure this tiny effect with the European Extremely Large Telescope and the Square Kilometer Array over the course of a decade or so.
In particular, a paper by Bolejko et al. (2019), entitled "Direct detection of the cosmic expansion: the redshift drift and the flux drift", demonstrates how the effect can be used to measure the expansion directly and distinguish between various cosmologies.