= Solution
The least index is $\boxed{\alpha=\omega_1}$, with
$$
\boxed{\sigma_{\omega_1}=\aleph_{\omega_1},\qquad
\operatorname{cf}(\sigma_{\omega_1})=\aleph_1}.
$$
Indeed, for every nonzero <limit ordinal> $\beta$, continuity of the <aleph number> enumeration and the <cofinality of an increasing ordinal supremum> give
$$
\operatorname{cf}(\aleph_\beta)=\operatorname{cf}(\beta).
$$
If $\beta<\omega_1$ is a nonzero <limit ordinal>, this <cofinality> is $\omega$, so $\aleph_\beta$ is a <singular cardinal>. The other infinite <cardinal numbers> below $\aleph_{\omega_1}$ are successor-indexed or $\aleph_0$, and are <regular cardinals>. For a <successor cardinal> $\mu^+$, a cofinal sequence of length at most $\mu$ would express $\mu^+$ as a union of at most $\mu$ sets of size at most $\mu$, contradicting <infinite cardinal arithmetic>.
There are $\aleph_1$ nonzero countable <limit ordinals>, and each countable initial segment contains only countably many. In increasing order they therefore have <order type> $\omega_1$. These are precisely the indices of the uncountable <singular cardinals> below $\aleph_{\omega_1}$. Hence every $\sigma_\alpha$ with $\alpha<\omega_1$ has countable <cofinality>, and the next one is $\aleph_{\omega_1}$, whose <cofinality> is $\omega_1<\aleph_{\omega_1}$.
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