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?