Some examples of properties which imply second-countability (or, equivalently, metrizability) for separable regular spaces can be extracted from the work on the Normal Moore Space Conjecture.

Recall that a regular space $X$ is said to be a *Moore Space* if there is a sequence $\{\mathcal{C}_n: n \in \mathbb{N}\}$ of open covers of $X$ such that for every $x \in X$, the family $\{st(x, \mathcal{C}_n): n \in \mathbb{N}\}$ is a local base at $x$, where $st(x, \mathcal{C}_n)=\bigcup \{U: U \in \mathcal{C}_n \wedge x \in U \}$.

Jones proved that if $2^{\aleph_0} < 2^{\aleph_1}$ then every separable normal Moore space is metrizable.

The assumption $2^{\aleph_0} < 2^{\aleph_1}$ is consistent with and independent of the usual axioms of set theory.

Is the set-theoretic assumption needed? Zenor proved that if you take the Moore plane and replace the x-axis with a Q-set (that is, an uncountable subset of the reals whose every subset is a relative $G_\delta$), then you get a separable normal Moore space which is not metrizable. Since every second-countable space has at most $2^{\aleph_0}$ many $G_\delta$ subsets and every uncountable set has at least $2^{\aleph_1}$ many subsets, it is clear that if there is a $Q$-set then $2^{\aleph_0}=2^{\aleph_1}$.

Furthermore, Zenor proved that if there is no Q-set then every separable Normal Moore space is metrizable, so the non-existence of a Q-set is equivalent to metrizability for separable Normal Moore space.

Traylor proved that every metacompact normal separable Moore space is metrizable.