Rumors of layoffs suck. Let's be upfront about that. There are no platitudes that make it easy. I know. I've been there. Take all the downsides and issues from a layoff and then layer in lots of fear, uncertainty, and doubt. You don't know if or when something might happen, or even who might be impacted, but life goes on. Work goes on. So what do you do?
I'm not suggesting that you simply ignore rumors, social media, Blind, or news reports. That's just unrealistic, and probably impossible. Yes, the rumors will impact you. It will impact your productivity. It will impact your attitude towards yourself, your work, your co-workers, and your friends/family.
What I have found that works for me, for teams I've worked on, and teams that have worked for me, is professionalism and staying focused. Regardless of the rumors, we all still have our jobs. The tasks that were there yesterday are still there now, and ignoring them won't make them go away. In all cases ignoring the work makes things worse overall, and in most cases ignoring it makes it worse for you personally.
Consider the best case scenario. It was just a rumor and nothing happened. If you had stopped working then you're just that much behind and now you and your team need to make up that lost time.
What if there were some layoffs, but you still have your job. Again, the work hasn't gone away. There will be some kind of schedule adjustment, but the amount of work is the same, just fewer people, so lost time is even harder to make up. How do you want to be thought of in 6 months or a year? How you respond now helps define how you are seen in the future.
And what if the rumors were true and you are impacted. I don't know about you, but I like my coworkers, and I want them to have a positive impression of me. It's highly likely we'll come in contact again, and I don't want to be "that person". Whether it's getting stuff done, sharing as much knowledge as possible, or just listening, it's preparing for my future. And who knows, you might want to work for the company again in the future. You don't want to burn any bridges.
But what about preparing for the future? Like I said, don't ignore the rumors. This is probably a good time to touch up your LinkedIn page and make sure your resume is up to date. Think about what's important to you in your job and your career. You should be doing that anyway, but this is a good reminder. Regardless of how this ends up you'll be having discussions about your career with your boss, so knowing what you want always helps.