When it comes to wildlife I tend to get better results when I shoot in cloudy overcast days. Back in November I did an early morning shoot in Richmond park which is home to hundreds of deer. In my bag, I had a Nikon D800, and my Sigma 150-600mm contemporary lens. Since it was pretty dull, I wasn't feeling too optimistic about getting good results. In these conditions, you could be tempted to use lower shutter speeds in order to shoot at lower ISOs but having wasted a lot of shots sue to camera shake, I wasn't having any of that. I had no monopod or tripod, so the bottom line was to eliminate shake by setting the camera to shutter priority. Most of my shots were at 600mm or close to it, so I fixed my shutter speed at 1/640 to 1/1000 sec. Having the shutter fixed I then had the camera set to auto-ISO in order to avoid under-exposure.
I spent almost a full day shooting and I must say, the high shutter speed worked very well. Despite the fact the ISO was set to 6400 at times, the shots were still very usable and the overwhelming majority were pin sharp. The moral of the story; ..... when shooting wildlife with a long zoom, the bottom line is you've got to have your shutter speed equal or higher than your focal length. I personally think it's best to get it closer to twice as much. So at 600mm I would suggest at least 1/800 but where you have enough light, I would suggest 1/1000 or 1/1200sec. Remember this is just to eliminate handshake not to eliminate shake from your subject. If you're shooting flying birds or insects then you need to go higher still with your shutter speeds.
Happy shooting folks!