This "book" might interest you:
I haven't found a book title or an author inside the book, but on page 35 they claim, that even calculating the eigenvalues/eigenforms for the non-flat 2-torus of the laplacian on 1-forms is basically impossible. I don't know whether that is something one would expect.
Unfortunately all I have is the above link, as I only got it from another thread:
But maybe you will find other information inside the book.
They have a "Hodge Theorem 1.30 for the Laplacian on k-forms" on page 34 that says:
For a closed connected orientable Riemannian manifold the eigenvalues of the Laplace operator on k-forms are all non-negative. There exists an orthonormal basis of $L^2(\Omega^k(M))$ consisting of eigenfunctions (or lets call them eigenforms I guess) of the Laplacian on k-forms. The eigenspaces are all finite-dimensional and the eigenvalues accumulate only at infinity.
So the eigenspaces of the Laplacian on k-forms are finite dimensional. Is that common knowledge?
I haven't read much more of "the book".
Somewhere else I read that the spectrum of the Laplacian only consists of its eigenvalues (common knowledge?): https://arxiv.org/pdf/1710.09579.pdf
on page 11.
This short text might also interest you:
starting on page 8, he explains that one can build up Morse-homology via a deformed Laplacian, where critical points of index k of a morse function correspond to eigenforms of the Laplacian on k-forms (the same k) and the exterior differential d on the eigenforms corresponds to the dual of the connecting-flow-lines counting boundary operator in Morse Homology.