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Part of the problem is lack of precise probabilistic model for this question (i.e., what is a "random" singularity?). You also have to restrict to isolated hypersurface singularities in ${\mathbb C}^n$. The question starts to make sense if you further restrict to Brieskorn singularities of the form $$z_0^{a_0}+...+ z_n^{a_n}=0$$ where all $a_k\ge 2$ are integers. This situation was analyzed in great detail, for instance, in Milnor's book "Singular points of complex hypersurfaces," sections 8 and 9. (See also Hirzebruch's paper accessible here.) For isolated singularities, provided that $n\ge 3$, the link $K$ of the singularity is a topological sphere $\iff$ it is an (integer) homology sphere $\iff \Delta(1)=\pm 1$ (theorem 8.5 in Milnor's book). Furthermore, say, for odd $n$, one can decide if $K$ is the exotic or not by looking at $\Delta(-1)$ mod $8$. For Brieskorn singularities, $\Delta$ is quite computable, so you can get your hands dirty and start computing probabilities. It is also, probably, easier, to use for this computation the graph-theoretic criterion on page 18 of Hirzebruch's paper.

I looked at the case when $n=2$ and the relevant question is about the probability of $K$ to be an integer homology sphere, equivalently, the numbers $a_0, a_1, a_2$ being pairwise coprime. Such probabilities were computed here and the answer is approximately $0.286$.

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Part of the problem is lack of precise probabilistic model for this question (i.e., what is a "random" singularity?). You also have to restrict to isolated hypersurface singularities in ${\mathbb C}^n$. The question starts to make sense if you further restrict to Brieskorn singularities of the form $$z_0^{a_0}+...+ z_n^{a_n}=0$$ where all $a_k\ge 2$ are integers. This situation was analyzed in great detail, for instance, in Milnor's book "Singular points of complex hypersurfaces," sections 8 and 9. (See also Hirzebruch's paper accessible here.) For isolated singularities, provided that $n\ge 3$, the link $K$ of the singularity is a topological sphere $\iff$ it is an (integer) homology sphere $\iff \Delta(1)=\pm 1$ (theorem 8.5 in Milnor's book). Furthermore, say, for odd $n$, one can decide if $K$ is the exotic or not by looking at $\Delta(-1)$ mod $8$. For Brieskorn singularities, $\Delta$ is quite computable, so you can get your hands dirty and start computing probabilities. It is also, probably, easier, to use for this computation the graph-theoretic criterion on page 18 of Hirzebruch's paper.

I looked at the setting $n=2$ when one would ask for the probability of getting an integer homology 3-sphere. Then the question becomes: "What is the probability for three natural numbers to be coprime," and the answer is $\frac{1}{\zeta(3)}$.

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Part of the problem is lack of precise probabilistic model for this question (i.e., what is a "random" singularity?). You also have to restrict to isolated hypersurface singularities in ${\mathbb C}^n$. The question starts to make sense if you further restrict to Brieskorn singularities of the form $$z_0^{a_0}+...+ z_n^{a_n}=0$$ where all $a_k\ge 2$ are integers. This situation was analyzed in great detail, for instance, in Milnor's book "Singular points of complex hypersurfaces," sections 8 and 9. (See also Hirzebruch's paper accessible here.) For isolated singularities, provided that $n\ge 3$, the link $K$ of the singularity is a topological sphere $\iff$ it is an (integer) homology sphere $\iff \Delta(1)=\pm 1$ (theorem 8.5 in Milnor's book). Furthermore, say, for odd $n$, one can decide if $K$ is the exotic or not by looking at $\Delta(-1)$ mod $8$. For Brieskorn singularities, $\Delta$ is quite computable, so you can get your hands dirty and start computing probabilities. It is also, probably, easier, to use for this computation the graph-theoretic criterion on page 18 of Hirzebruch's paper.

I looked at the setting $n=2$ when one would ask for the probability of getting an integer homology 3-sphere. Then the question becomes: "What is the probability for three natural numbers to be coprime," and the answer is $\frac{1}{\zeta(3)}$.