at.algebraic topology - circle action on sphere

I do not know, in which category your actions live. In the following answer I want to consider isometric actions.

Another classification problem might be:

Classify all periodic one parameter subgroups of $SO(n)$ up to conjugacy, which is the same problem as the classification of all such actions up to isometry.

There is a normal form for orthogonal matrices over the reals, which says, that every such matrix is conjugated to a block matrix consisting of $(2,2)$ rotation matrices and $(1,1)$ diagonal entries, which are $pm 1$.

My claim would be that every isometric action decomposes (after congugation) uniquely as a induced action on each of the components of the decomposition

$S^n = S^1 * S^1 * ldots * S^1 * S^0 * ldots * S^0$
($*$ should denote the join here). The isometric actions of $S^1$ on $S^1$ can be classified by the integers. The only possibilities are $(t,x)mapsto mt+x$ for $minmathbb{Z}$.

I think the upper normal forms for matrices should imply this. But I am not sure.