Usually you won't need an application server, even though there's really nothing that stops you from doing it. Spring handles most of the stuff you would normally use an application server to do; object management, transactions, persistance, plus a whole lot of other useful stuff.
For your average webapp you just use
Tomcat, setup a Spring context in the configuration files and you're on your way.
J2EE services, such as JMS server, JNDI server and EJB container, are not included in Spring (although it helps you to integrate against the services). You usually don't need the two latter if you're building your system from scratch, and if you need to use JMS you can use a standalone JMS provider, so in my opinion there's very few situations where you need a full application server.