I need to create a social-networking application that is going to attract quite a significant number of users for its use and is going to require high scalability capabilities. I would prefer an open source solution and had MySQL in mind. But I need to know if MySQL will give me what I am looking for.
Yes.
You can extend MySQL with Replication to a large number of servers with redundancy, failover stuff and good performance.
MySQL will cope with heavy usage.
More importantly your system should be structured in such a way that you place the minimum load on the database and so that you could change the database if needed.
Both of the big OSS DB's (MySQL and PostgreSQL) can scale to extremely large sites. The trick would be to plan from the start such that data can be broken up ("shard") into multiple DB servers and you can quickly find out which server you need to go to.
For example: users having nickname A-G go to server1, H-O to server2, etc. This way you can scale almost infinitely because when a server reaches the limit, you can just split it into smaller pieces.