To make such a web application work, it would need (in the US) to use the "safe harbor" provisions of the DMCA (17 USC 512(c)). (If not in the US, it would need to make use if the comparable law in that country. Most nations now have more or less comparable laws, because the modified Berne Copyright treaty required some implementing law.)
This would mean that the provider of the app, which would be a form of information service, would need to accept and act on DMCA take-down notices. When a proper notice by a copyright holder or a holder's authorized agent was submitted, the service would have to investigate, and if it seemed likely that the content specified in the notice was infringing, remove it from the site or disable access to it. The provider would need to notify the user who had uploaded the content that it was being taken down.
There are many places on the web which describe in detail the process of receiving and responding to DMCA takedown notices. I will not go into detail here.
YouTube's copyright procedures are an example of the sort of thing I am talking about. If a web site allowing user upload of potentially copyrighted content, such as music, follows this sort of procedure, it should be safe from suits for infringement based on content uploaded by users. Whether implementing such procedures is too costly and troublesome is a decision anyone contemplating such a site would need to make.