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My favorite reference for point-free topology is the very new book.

Frames and Locales: Topology Without Points by Picado and Pultr.

This book is an excellent book for those who want to learn about point-free topology for the first time and as a reference for those who are already familiar with point-free topology.

As for recent results in point-free topology, I have recently been researching a duality in point-free topology. My new duality represents all zero-dimensional frames as Boolean algebras along with specified least upper bounds.

We therefore define a Boolean admissibility system to be a pair $(B,\mathcal{A})$ such that $\mathcal{A}$ is a subset of the powerset $P(B)$ that satisfies the following properties.

1. If $R\in\mathcal{A}$, then $R$ has a least upper bound.

2. $\mathcal{A}$ contains each finite subset of $P(B)$

3. If $R\in\mathcal{A},S\subseteq B,S\subseteq\downarrow\bigvee R=\{a\in B|a\leq\bigvee R\}$ and $R$ refines $S$(i.e. for each $r\in R$ there is an $s\in S$ with $r\leq s$), then $S\in\mathcal{A}$ as well.

4. If $R\in\mathcal{A}$ and $R_{r}\in\mathcal{A},\bigvee R_{r}=r$ for $r\in R$, then $\bigcup_{r\in R}R_{r}\in\mathcal{A}$

5. If $R\in\mathcal{A}$, then $\{r\wedge a|r\in R\}\in\mathcal{A}$ for each $a\in B$.

Property $1$ states that $\mathcal{A}$ is a collection of least upper bounds and properties $2-5$ state that $\mathcal{A}$ contains all sets with least upper bounds that you would want to include. For instance, in a Boolean algebra you would always want to include the least upper bound of a finite set. Axioms $2-5$ get rid of all the trivial differences between Boolean admissibility systems. A Boolean admissibility system $(B,\mathcal{A})$ is called subcomplete if whenever $R\cup S\in\mathcal{A}$ and $r\wedge s=0$ whenever $r\in R,s\in S$, then $\bigvee R$ exists.

I recently proved that the category of Boolean admissibility systems is equivalent to the category of all pairs $(L,A)$ such that $L$ is a frame and $A$ is a Boolean sublattice of $L$ which is a "basis" for $L$(i.e. $A$ is a sublattice of $L$ consisting of complemented elements where each element in $L$ is the join of elements in $A$). This equivalence of categories restricts to an equivalence between the category of all zero-dimensional frames and subcomplete Boolean admissibility systems.

With this duality, I was able to characterize point-free topological properties in terms of the corresponding Boolean admissibility systems. These properties include ultraparacompactness, ultranormality, $\kappa$-compact zero-dimensional frames(where $\kappa$ is a cardinal), extremally disconnected frames(as Boolean admissibility systems which are complete Boolean algebras), Lindelof $P$-frames(as $\sigma$-complete Boolean algebras), and other properties.

This result does not have as much of a pointed analogue since very rarely does a Boolean admissibility system correspond to zero-dimensional space (i.e. a spatial zero dimensional frame). The Boolean admissibility systems that correspond to topologies are precisely the subcomplete Boolean admissibility systems $(B,\mathcal{A})$ where each ideal closed under taking least upper bounds in $\mathcal{A}$ can be extended to a maximal ideal closed under taking least upper bounds in $\mathcal{A}$. This property can be characterized by a very strong distributivity property and very few Boolean admissibility systems satisfy this property.

I should also note that one can represent any pair $(L,A)$ where $L$ is a frame and $A$ is a "basis" for $L$ as the poset $A$ along with specified least upper bounds. Unfortunately, even though this setting is more general, I have not yet found a way to represent any separation axioms in terms of posets with specified least upper bounds.
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