I'll take a crack at it. Replace numbers, dots, dashes, and spaces with nothing, assuming all files are mp3's. (Or "copy-item -destination")
# test files
# echo hi | set-content '02. SongName.Mp3','02 - OtherName.Mp3','02 MoreNames.Mp3','03. Song.mp3','03 - Songs.mp3'
dir *.mp3 | rename-item -newname { ($_.basename -replace '[\d\.\- ]') + '.mp3' } -whatif
What if: Performing the operation "Rename File" on target "Item: /Users/js/foo/02 - OtherName.Mp3 Destination: /Users/js/foo/OtherName.mp3".
What if: Performing the operation "Rename File" on target "Item: /Users/js/foo/02 MoreNames.Mp3 Destination: /Users/js/foo/MoreNames.mp3".
What if: Performing the operation "Rename File" on target "Item: /Users/js/foo/02. SongName.Mp3 Destination: /Users/js/foo/SongName.mp3".
What if: Performing the operation "Rename File" on target "Item: /Users/js/foo/03 - Songs.mp3 Destination: /Users/js/foo/Songs.mp3".
What if: Performing the operation "Rename File" on target "Item: /Users/js/foo/03. Song.mp3 Destination: /Users/js/foo/Song.mp3".