Probably the interesting examples involve large cardinals.

Instead of just collapsing $\mathbb R$ to be countable, force with $Col(\omega,<\kappa)$, where $\kappa$ is an inaccessible cardinal. This is the direct limit of repeatedly making the reals countable. So for each ordinal $\alpha \leq \kappa$, there is a new version of the reals $\mathbb R_\alpha$, and for $\alpha < \kappa$ these all become countable in the end. This feature is essential to the landmark theorem of Solovay that in the final model, every definable set of reals--definable meaning is in $L(\mathbb R)$--is Lebesgue measurable, equal to an open set modulo a meager set, and if uncountable contains a perfect subset. The proof is presented in Chapter 26 of **Set Theory** by Thomas Jech. It is a bit involved, but a key step that relates to your question is the following. Suppose $\mathbb R$ is the version of the reals in the final model, and $r \in \mathbb R$. Then in the final model, the intersection of all measure one Borel sets coded in $L(r)$ is measure one in $L(\mathbb R)$. This is because in the final model there are only countably many of these.

Now with larger cardinals this enables us to actually say things about "our own universe." If there is a Woodin cardinal $\delta$, then there is a forcing called the stationary tower that turns $\delta$ into $\aleph_1$ and has some special features. Say we start in the model $V$ and $G$ is the generic filter for the stationary tower. Then:

(1) There is a submodel $M$ of $V[G]$ that is closed under countable subsequences (thus contains all the new reals), and an elementary embedding $j : V \to M$.

(2) There is another forcing extension $V[H]$ with the same reals as $V[G]$, where $H$ is generic for $Col(\omega,<\delta)$.

Thus Solovay's theorem may be applied to say that every set of reals in $L(\mathbb R)^{V[G]}$ has all the regularity properties. By the elementarity of $j$, this holds for $L(\mathbb R)^V$ as well.

There are set theorists on MO who are far more expert than me on these topics, so probably even more fascinating results can be given.