'Eigenvectors' of evolute operation

Light studied the problem, but didn't solve it in his 1917 dissertation. A 1920 note of Light was followed by a 1921 note of Franklin showing the existence of infinitely many classes of examples not covered by Light, and pointing out that Puiseux had shown this in 1844. However, the question of determining all of them isn't addressed there, and I don't know what else is known.

Regarding your question about formalizing the notion of curves as eigenvectors, couldn't you just take the real vector space whose basis consists of representatives from each similarity class of curves? Scalar multiplication of a basis element by a real number corresponds to dilating or shrinking, and reflecting if negative. The evolute operator is formally extended to this vector space linearly, and the curves similar to their evolutes will be precisely (scalar multiples of) the basis vectors that are eigenvectors.

All these curves are "multihedgehogs" : for clear definitions and neat examples see

Y. Martinez-Maure, A Sturm-type comparison theorem by a geometric study of plane multihedgehogs, Illinois Journal of Mathematics 52 (2008), 981-993

or

Y. Martinez-Maure, Les multihérissons et le théorème de Sturm-Hurwitz, Archiv der Mathematik 80, 2003, p. 79-86.

Such a curve can be defined by a 2Nπ-periodic function of class $C^2$ on $R$, (the number $N$ is merely the number of full rotations of the coorienting normal vector $u (t) = (cos t, sin t)$. Its evolute as support function $h'(t-π/2)$. When you say that the curve is an "eigenvector", that means in fact that its support function is a spherical harmonic that is, an eigenfunction of the circular Laplacian.

There are of course extensions of these notions in higher dimensions.