Can an iPhone 5S play music that's stored on a USB drive? My wife's iPhone 5S doesn't have enough storage space for all of her music.  If we were to buy a USB drive that connects to her phone, could we put her music on that and then have the phone play it from there in a reasonably hassle-free way?  What would the process be?
You can read a flash from an iOS device, but this capability is limited to some very fancy lightning-port enabled USB flash drives. 

For most use cases, I would recommend using a cloud service like Google Drive if you're looking to listen to purchased music on an iPhone without using up storage space. Essentially, you can just drag your music purchased from anywhere into Google Drive, download the app onto your iPhone, sign into your account, and listen to your music (though you of course need an internet connection unless you download them to your phone). 

However, if you want to buy a special flash drive, you can purchase the Sandisk iXpand Flash Drive, which plugs directly into the lightning port and can play music while plugged into the iPhone, through the Sandisk iXpand app. Just connect it to a Mac/PC beforehand to load music onto it through the standard USB port.



tl;dr: Buy this flash drive or use Google Drive.



Edit: don't forget about streaming services like Spotify and Apple Music, which don't take up storage space either.