Note that, in knowledge representation and reasoning, common-sense knowledge is traditionally represented as sentences (in logic). For example, one possible sentence that you could store in a knowledge base is
The earth revolves around the sun.
This is common-sense knowledge (ignoring the fact that this was not the case in the past until Copernicus and Galileo came along in the 1500s). The programming language PROLOG is based on this type of knowledge, facts, and deduction.
Self-supervised learning (SSL) has been used to learn representations of data (which is often done in the context of natural language processing), but these representations may not be knowledge, in the sense that we may not understand what is encoded in these representations or whether they are related to our common-sense knowledge.
So, whether or not SSL can be used for acquiring/approximating knowledge depends on the definition of knowledge that you use.
If you use the traditional definition of common-sense knowledge, SSL is not usually used for knowledge representation, but it has been used in the context of knowledge graphs (see the SelfLinKG approach), which can be viewed as a graphical representation of a knowledge base. So, it's possible that SSL can also be useful for approximating common-sense knowledge.
To answer your question more directly, my impression is that knowledge graphs are promising and they are useful in practice. Google uses them in its search engine (and probably also in the Google Assistant). Whenever you search, for example, for a famous person (for example, Gandhi), on the right side, you should see a window describing certain details of that famous person. That's done using Google's knowledge graph.
Right now, people are trying to develop techniques to learn embeddings of the nodes or relations in a knowledge graph, with the goal of using these embeddings for discovering new knowledge. This area is called knowledge graph embedding (KGE). My other answer provides more details (but there are many surveys on KGEs that you can find, for instance, on Google Scholar).
Having said this, knowledge representation has been a big problem since the early days of AI (for example, see the frame problem) and there hasn't been much progress in this area (AFAIK). Most people are focusing on neural networks right now, but explainable AI techniques for neural networks could potentially play a role in knowledge representation.