There have been some discussions before and opinions differ.
I've been shooting at 1/60 for my Nikon Coolpix 5700 at 8x optical zoom (actual 71mm which is 280mm on the 35mm equivalent format) without any much handshake blur problem.
An image formed using 71mm (280mm on 35mm equivalent) on the Nikon 5700 is about the same size as one formed using 71mm (about 420mm on 35mm equivalent) on the FZ7. The difference in 35mm equivalent is due solely to the difference in image(CCD) sensor size. The Nikon 5700 has a larger sensor and therefore the subject appears smaller in the frame than the same image in the frame taken by the FZ7 and therefore the subject looks as if it's taken at only 280mm on the 35mm equivalent format compared to 420mm on a 35mm equivalent format by the FZ7 even though the actual subject size in the image formed is the same for both pictures taken at the same actual focal length (71mm in this example).
Indirectly, it also means that both images are susceptible to the same handshake blur risk if they're taken at the same actual focal length. The 35mm equivalent focal length is just a number telling us the focal length that would be used to have the subject(s) appear this size in the frame if the film/CCD sensor is the full frame 35mm format.
The picture below was taken at 71mm (280mm equivalent on the 35mm format on my Nikon 5700) at 1/60 shutter speed and there is no VR (Vibration Reduction for Nikon, equivalent to OIS for panasonic or IS for Canon) and it's near the threshold of getting handshake blur. If the 35mm equivalent focal length is erroneously used as a guide, then it would demand shooting at least 1/280 already (check the EXIF info in the picture) :