I’m confused by your choice of three terms to compare. Within RabbitMQ, Fanout and Direct are exchange types. Pub-Sub is a generic messaging pattern but not an exchange type. And you didn’t even mention the 3rd and most important Exchange type, namely Topic. In fact, you can implement Fanout behavior on a Topic exchange just by declaring multiple queues with the same binding key. And you can define Direct behavior on a Topic exchange by declaring a Queue with * as the wildcard binding key.
Pub-Sub is generally understood as a pattern in which an application publishes messages which are consumed by several subscribers.
With RabbitMQ/AMQP it is important to remember that messages are always published to exchanges. Then exchanges route to queues. And queues deliver messages to subscribers. The behavior of the exchange is important. In Topic exchanges, the routing key from the publisher is matched up to the binding key from the subscriber in order to make the routing decision. Binding keys can have wildcard patterns which further influences the routing decision. More complicated routing can be done based on the content of message headers using a headers exchange type
RabbitMQ doesn’t guarantee delivery of messages but you can get guaranteed delivery by choosing the right options(delivery mode = 2 for persistent msgs), and declaring exchanges and queues in advance of running your application so that messages are not discarded.