Rumors are the product of a vacuum. And they don't start as rumors, they start as conjecture. Staff are smart, they think about what may be going on and consider the possibilities. Sometimes it's because they are trying to understand what the impacts may be on the company or even themselves.
If there are rumors circulating around your team about your department or the company, it's really a failure of management to understand their staff and a failure to communicate.
It's natural and understandable for staff to think about and talk about issues that are affecting their company and may affect their jobs. To dismiss rumors as childish or to try to try to banish them from the company or your team, and in particular to try and tell people to stop is to misunderstand the underlying issues that create the rumors in the first place. Of course, I'm not talking about personal rumors about people that shouldn't be anybody else's business. I'm talking about rumors related to the company or organization itself.
Anytime there is change or potential change in the company, or even something that looks like change, people will think about what the impact is to them. And they may share their thoughts with their colleagues to see what they think. A perceptive Manager understands this and recognizes what is happening. They will manage their activities and communications differently as a result.
Even just a little bit of information can cause rumors to happen and be passed around through the grapevine. It may be something inadvertently leaked, an event, a flurry of activity, out-of-town visit visitors and even financial types, lawyers, out of town executives or others suddenly visiting frequently will raise questions.
And the more you try to suppress rumors, the more you will create along with a very negative impression. While it's clearly not possible to reveal some information that may be the root cause of the rumors, acknowledging that something is happening by either providing some sort of timeline or general context will help. Clearly and honestly ruling out possibilities that might be cause for worry by your staff should be done if possible and you should clear the air as soon as possible, even advancing activities and a formal announcement if possible when the type and nature of rumors, and the possible ill-will or concern it generates, warrant it.
Of course that won't stop the conjecture from continuing, but you will have created more goodwill and a more positive impression of the company and its management with staff until the truth can be shared with everyone.
The sense of the unknown worries everybody. By providing some information and possibly identifying a timeline for more information, you will help to dampen down the negative energy that's created by the rumor mill. Just be sure to follow-through on your timeline and provide updates if you can't.