As a language how stable is Groovy? Do any big names use it?
As for "big names", there may be some (you shouldn't care). Obviously, nevertheless, Groovy's got a basic problem with its poor performance compared to Java. Positively, it's doubtable that Groovy is used in any mission-critical projects where scalability counts.
Regarding stability (as in the "absence of bugs"), take a look at the Groovy issue tracker. Enjoy ROFL'ing at several bugs where Groovy (again and again) has stumbled upon its own "magic". - Groovy bugs are numerous, and often unresolved for years.
Regarding stability again (as in "backwards compatibility"), I'm too new to Groovy to say much about that. Nevertheless, the closures syntax had changed without a compatibility layer provided. (This would never ever happen in the Java language.)
- When looking at those points from a Java perspective, Groovy is a kindergarten. From a Scala perspective even, Groovy will never grow up.
- Which is not to say that Groovy had no clear advantages in other scopes (like, there is Grails). But you've asked about these.