Let $M^2$ be a closed surface, say the 2-sphere. Is there any example of metric on it such that there are uncountably many points are conic and the metric is smooth elsewhere?

We call $p\in M$ a conic point if there exists $\lambda_i\to \infty$ such that $(\lambda_i M, p)$ converge to a linear cone $C(S(\ell))$, the cone over a circle of length $\ell$, for $\ell\ne 2\pi$.

For convex surface, there at most countably many conic points, which is proven by AD Alexandrov.