You can sleep safe - there is a no strange readings on the map now.
But one day you can wake up at the morning and see for example 20msV or other "big" value.
What I'm should to do? - It's a first question.
Definitely first is a check neighbours reading on the map and other sources of information (some countries have radiation monitoring systems available for public), press, radio and Internet is a important source of information as well. If you somehow confirm you have no faulty sensor (other sensor near you have as well unusual reading) you have to decide to act somehow.

Most of us heard critical dose for person is a 100 Rad (for ~3 month). All doses above this value are seriously dangerous for your health and can be lethal.

What you do? How your action depend of reading if will be for example 10, 50, 100 micro sV - or more?

If you get a large/lethal dose, you can start by taking an iodine supplement, straight out. This said I wouldn't go running for the iodine, if the reading is suspect, out of the blue, and doesn't jive with anything to account (true outlier). For such statistical flukes in the data point, I'm sure the project has ways to account, when collating the results from multiple sources; but for your own benefit...

As to yourself, if there is some known cause (Fukashima disaster for instance), and then you're seeing such a reading, you might want to grab an iodine supplement at that point.

From there, depending where you are, probably contact your doctor, the emergency room, head to the hospital, etc. If you have the data on hand, someone in the medical profession should know how to react, should there be something there. That said, I wouldn't suggest panicking (that usually doesn't help much of anything, as many emergency responders will know full well from having to deal with crisis situations on a daily basis), and one also doesn't want to become a hypochondriac of course. There could well be something with a placement in one's house or whatever, that might be giving a false reading.

On the other hand, in the US for instance (not sure where you live though), we have had some radiation events, such as Three Mile Island, who's reactor isn't all that far from me in New Jersey, being around Harrisburg (I believe), PA. About a 35 min drive from center city, Philadelphia here, so I'd estimate a 200 miles (give or take), as Harrisburgh is of course west of Phila, and I'm just northeast of it. Naturally though the reactor that entered meltdown has been dis-used, though the other reactor (Three Mile Island had 2 reactors) is still in use.

On the other hand, sometimes the information does get confused, as happened in Japan when TEPCO was giving information that tended to downplay the incident, with nuclear officials in the US suggesting that it sounded like a much more serious matter was taking place there, and the citizens of Japan left in confusion as they were getting conflicting stories from different sources, and TEPCO arguably reacting somewhat from the position of public relations/damage control. The IAEA got to the bottom of it as time progressed, and they came to inspect/investigate, but of course where the biologic consequence is concerned, people don't have years to react.

Some judgment does have to be exercised, along with, if you're getting a reading "does this make sense". How one would respond to a strange spike, if they lived near a nuclear reactor, would also differ from how they would react if there was no such source to explain nearby. Not all radiation is man made either, there are a lot of natural sources, some of which even come from space, though our atmosphere filters much of that out...

At the time of Chernobyl disaster I was quite young and was living in Poland, 300 miles away from this explosion. Our government apply Iodine ( Iodi solutio aquosa) to all children. I have lucky to one from my teachers was nuclear specialist, so he confirm this is a good solution (not all belived it is ok for us!).
From that time I remember there was a lot of misconducting informations about problems, doses and danger level. Definitely better is a to collect this info and now "what to do" before SHTF. Of course it is hard to predict what will happened - if there will be any explosion, attack, will have local or global range. Moreover - if we will be in EMP (electromagnetic pulse) range our access to internet information can be dramatically reduced. So in my opinion better is a be prepared BEFORE something happens.

Yeah, and what made things worse there, was that the Soviets had pretty much run Chernobyl without any shielding... Was essentially a nuclear reactor, in an open room type scenario. Definitely something we know, not to do, for obvious reasons...

Of course state side we have also had accidents, Three Mile Island for instance, where one of the reactors went critical (there were 2), and the other is still functional to this day. Nuclear reactor technology has improved over the decades, but with the uncertainty being what it is for many, the politics has tended to be towards stopping production on new reactors, though the older ones, still grandfathered in, remain in operation. And without a replacement of some form, sometimes beyond their original projected life span. The irony of course is, that the 40 year old reactor really is less safe, then one built today, would be.

Of course we're dealing with radiation, so people shouldn't take that to mean, no risk at all, but sometimes in the too and fro, many don't always understand that risk, and react without understanding. Of course a certain amount of radiation, is in the environment, even we're radioactive to a small degree, as a small percentage of the carbon in our body would be C-14, though most would be C-12. This is of course how archaeologists are able to do radio carbon dating on artifacts, as well as skeletal remains, to try to get a relative dating for various dig sites... That of course also has it's matters to be adjusted to, as both the burning of fossil fuels (coal was in the Earth from before the time of the dinosaurs, and even dating back to the time of the Permian extinction), as well as radioactive explosions such as occurred at Hiroshima (for obvious reasons) can effect the baseline, they're testing things against, to get their dating measure, relative to the half life of C-14, and how much is left in the artifact now, vs in the environment at large, today... Once the coal for instance, which was in the Earth for 100s of millions of years is burned, the C-14 from it would have had a lot of half lives, but now burned and turned into CO2, that carbon is reintroduced into the ecosystem, where it would then form plants, and enter the food chain as such ;)

What happened during the Fukashima disaster though, shows exactly how the information getting out can go though. Tepco, (which to a certain degree was in damage control mode, as they responded with their own PR) was saying one thing. The Japanese government was largely getting the information from Tepco, and nuclear authorities in the US were saying something quite different, when assessing the risk. Then the IAEA had to come in, to assess the situation, but the people on the ground, in Japan were themselves confused wrt who to believe, and what they should do NOW...

All one can do though, is respond to the information they will have at hand though, and respond in the best way they are able. Of course reason I mention considering the risk or one's area, is because if someone gets a fluke reading, they could end up with a reading of a Chernobyl like disaster in Antarctica (if one were to go down there, for who knows what reason), and that just wouldn't fit. We don't have nuclear reactors at the south pole obviously, and I couldn't see any reason one would use a nuke on the polar bears :o Maybe above, if an asteroid were hurtling towards Earth, and they were trying to change it's trajectory, but if they were going to attempt that, they also wouldn't necessarily be trying it so close to the Earth's surface. Our telescopes can tell us when one of these things is coming, long before it comes that close. We're tracking one, for more then a decade away, but it's expected to miss us, after the math was run, though it will come closer to Earth then the moon, like one recently had.