I have constructed a list of surgery coefficients which yield spherical space forms. For instance, there are two knots with different Alexander polynomials on which 29-surgery will give a small Seifert fibered space of $S^2(2,3,5)$-type. Now I am just wondering does anyone know any knot on which 58 or 83 surgery will give a spherical space form. I would appreciate any help very much.
Lens spaces of orders 58 and 83 may be obtained by integral surgeries on torus knot; see Moser's "Elementary surgeries on torus knots" which gives the classification up to homeomorphism. (Mind Moser's conventions.)
58/1 surgery on the (3,19)-torus knot yields the lens space L(58,-9), and
83/1 surgery on the (2,41)-torus knot yields the lens space L(83,-4).
Here I'm using the convention that -p/q surgery on the unknot yields the lens space L(p,q) so that +5/1 surgery on (2,3)-torus knot yields L(5,-4).
Edit: Of course any non-zero surgery on unknot is spherical, but I reckon you were looking for something less trivial.