Aerospike nose cones for supersonic jets? Not to be confused with aerospike nozzles. This is about the big pin-like rod in front of ICBM nose cones.

This answer (from Space) about aerospike nose cones notes that one of the potential advantages of the aerospike nose cone is to considerably decrease noise from the supersonic boom. In this comment the author evokes projections for the Concorde, dropping noise from 110 dB to 60-70 dB, that is 15 to 30 times quieter.

Noise from the supersonic boom, as the story goes, is what ultimately killed the Concorde: most nations forbid overland flight because of it, limiting it to ocean flights. So there would be clearly benefits from this.

Have there been any supersonic planes using this design that flew? Are there projects to do so that are currently active? Otherwise, why not?
There are no historical aircraft with an aerospike nose, at least, none that I remember, and none that I could find after 30 mins of research. I'm not sure if there are any planned projects using it, but I doubt there are, given the purpose of aerospike noses - aerospike noses really only have one main purpose - to streamline a blunt body. This is commonly used in missiles and such (especially in SLBMs), where height/length is a constraint - a slender, sharp nose is very long, so to save height, those missiles are fitted with an aerospike that telescopes out. The flat plate at the end of the aerospace forms a detached bow shock, which makes the airflow "see" the blunt body as having a different, more streamlined shape, reducing drag. Here's a visual of that happening:



Image source - https://iopscience.iop.org/article/10.1088/1742-6596/1276/1/012031/pdf

However, the bow shock generated by a drag reducing aerospike causes more drag and a heating than the oblique shock generated by a slender nose of the same length. For SLBMs, a long, sharp nose is obviously out of question (well, you could have a telescoping sharp nose, but that would be heavy), but for aircraft, length really isn't a concern, you can make the nose as long as needed. Therefore, it doesn't really makes sense for an aircraft to have an aerospike nose.