I too have been checking on the new fz35 my cousin bought. Though it is a very good camera overall, the noise problem is very much there. I haven't compared it to any other camera, but even my almost 4 year old Panasonic lz2 has significant noise.
The fz35 has noise right from iso100, and sometimes even in bright light. One solution is to disable the in camera noise reduction, and process your photos with Noiseware. Though minor details are lost, it is still a good alternative. Alternatively, use the manual or shutter priority mode, and set the shutter speed manually to around 1/250, basically on the higher end, that is slower shutter, depending on the light conditions. Accordingly, you can use the lowest iso. You can try to slightly overexpose photos and make up for it by increasing contrast or reducing brightness on the computer. We found the noise somehow increased as the zoom increased, though we didn't use digital zoom at all. I cannot explain that, but you can test this point for yourself.
Though reducing resolution can reduce some noise, I have not found significant difference. I haven't really tried night photos except the moon, and I was quite happy with that. Once I have tried the starry sky at 30sec and 60sec at min iso, and got good results.
But you can be happy since this camera still has competitive noise levels for its competition and is one of the finest superzooms available!
As for you, vitalyx,
From what I have read, a smaller sensor in general does mean more noise, but there is a lot of other technologies and variables involved. So, there is no saying that for sure. And if you are considering the fz35, there is only the Canon sx20is in real competition, which has its own problems. If you haven't purchased this camera already, try considering the Fuji s200exr. It has 14.3x optical zoom, and will take the noise part properly. Again, it has its own limitations like only 640*480 video capture, but image quality is very good. It has a special low light mode, which seems to be really good considering the reviews I read. If you are considering Night photography seriously, maybe this could as well be the best option for you. There is no review available at cameralabs, so you may have to Google it... If you are still considering the fz35, the noise will be there, little or too much, but it will be there. It wasn't a deal breaker at all for my cousin, so it just depends on how much is too much!