Music Library on external HD connected via USB

Yes, you can do this assuming the router supports SMB / CIFS file sharing from the USB port (most do).  You'll need to know two things, the router's local name (netbios name) and the shared folder path.  If you're not sure of where to find this information, consult the router's support documentation.

Setting up the music share with the shared drive on your router is similar to setting it up on the computer.

From the Manage menu, select Music Library Settings... shared music folders that are currently available to your Sonos Wireless HiFi System appear.  Click + on a Mac, or Add from a PC. 

To add the location from your router, select On a networked device such as a network attached storage (NAS) drive. 

• Type the network path for the music folder \\Name\Sharename, where Name is the network name for your router, and Sharename is the name of the top level shared folder, or click Browse to search for it.
• If it is not shared anonymously, enter the user name and password of a user with permission to access this folder, and click Next.
• Click OK to confirm.

Thanks for the reply, do I need to re-index my library every time I add new music?

Thanks for the reply, do i need to re-index my music library every time I add music to the HD

Yes, you do need to reindex after adding new music although you can also set your share to index automatically at a certain time every day (such as the early hours of the morning, when you won't necessarily be in front of the system) in case you forget.

You'll find this under Music Library settings in the Advanced tab.

Can the router's local name (netbios name) simply be its IP address?  My router says:

Storage Sharing function is based on NetBIOS/SMB protocol which is supported by most Windows operating system and some other operating systems. All active volume(s) will be shared and authentication is not required. You will be able to access the folders with the following methods: 
   For Windows OS:Open "Run" window in the Start menu and enter \\(IP Address) or \\(IP Address)\(Share Name) e.g. \\192.168.1.1 or \\192.168.1.1\photo;
    For Mac OS:Open "Connect to Server" window in the Go menu and enter smb://(IP Address) or smb://(IP Address)/(Share Name) e.g. smb://192.168.1.1 or smb://192.168.1.1/photo.

Can the router's local name (netbios name) simply be its IP address?  My router says:

Storage Sharing function is based on NetBIOS/SMB protocol which is supported by most Windows operating system and some other operating systems. All active volume(s) will be shared and authentication is not required. You will be able to access the folders with the following methods: 
   For Windows OS:Open "Run" window in the Start menu and enter \\(IP Address) or \\(IP Address)\(Share Name) e.g. \\192.168.1.1 or \\192.168.1.1\photo;
    For Mac OS:Open "Connect to Server" window in the Go menu and enter smb://(IP Address) or smb://(IP Address)/(Share Name) e.g. smb://192.168.1.1 or smb://192.168.1.1/photo.
This is what my router says too! Advice much appreciated!

what is the answer to these two?????????????????

The answer is "yes", the router's local name can simply be its IP address.

I am trying to setup a USB with music on it plugged into the ASUS RT-AC87U.
I want to add the files on the USB as a music folder on my SONOS.
The SONOS cant connect to it. I get
Reason The computer RT-ac87u-57f8 refused to let Sonon connect to it.
I have Allow guest login - ON.
Other PCs on my network can play the music from the USB connected to the ASUS RT-AC87U, without a username and password.

I am trying to setup a USB with music on it plugged into the ASUS RT-AC87U.
I want to add the files on the USB as a music folder on my SONOS.
The SONOS cant connect to it. I get
Reason The computer RT-ac87u-57f8 refused to let Sonon connect to it.
Sometimes Sonos can object if the host name contains anything other that alphanumeric characters or "_" (underscore). Your name contains dashes.

Try specifying the network location in Sonos Library Setup using the IP address, e.g. //192.168.1.1/share_name (substitute as appropriate).