Title. I want to be able to use apps like Spotify on my school’s network, but it’s blocked. Any VPNs for my whole system?
By the very nature of VPNs they are system wide, unless it is specifically made to be in the browser only. Proper VPNs would be setup at the network level and every thing network wise would go through it.
Do you mean free ones? Cause there are tons of paid VPNs that work system wide.
I use VPN Unlimited from KEEPSOLID. It works system wide, you just need to install the software and select the server to connect to. I have paid subscription, but the free version works well too, with less number of servers.
It also has Chrome extension, so if you just want it on browser, it can work like that.
Any VPNs for my whole system?
Depending on how competent your schools IT dept is you probably wont be able to do any admin stuff system wide. Or even if they allow VPN traffic through firewall. If I was the sysadmin there I wouldn’t allow it.
They only seem to offer a 7 day free trial now, and it is pretty expendive compared with other players.
Yeah, saw that now. On Apple App store, it’s 7 days. On Microsoft’s, it is 10 days. But any good VPN will cost money, otherwise “You’re the product!” 
Yeah - as my dear (departed) old gran used to say “you only get what you pay for!”