In principal, Facebook Messenger certainly is safe. The app itself simply allows a use to interact with their friends online, much like MSN Messenger did years ago.
Having said that, there are some concerns that Facebook Messenger takes advantage of people's personal information, and is therefore in breach of privacy. Personally, I've never seen an issue with this as I choose not to share personal information online that I wouldn't want someone else to see.
On the other hand, there are some issues with Facebook Messenger that I consider to be out of Facebook's control for the most part. Here are the two most obvious examples.
Speaking to people you actually know in the real world is a pretty safe thing to do on Facebook Messenger. Issues can arise, however, if you're speaking to complete strangers.
There have been numerous cases of fraudsters befriending people online, and then convincing them to send them money or other aid that they have no intention of giving back.
The other major safety problem I see with Facebook Messenger is the lack of control over who can use the application. Although you need to confirm that you're of suitable age to make use of the app, there isn't really anything in place to stop a child from lying in order to gain access.
Unfortunately, there are plenty of predators in the world that would love to speak to a child inappropriately, and they may be disguised as another child to make it difficult to tell at first glance.
The only solution is for parents to make sure that they know who their children are talking to online, and more preferably, control which applications their children have access to.