My camera (Nikon D5600) does not have in-body image stabilization. Even the VR kit lenses seems to be of no use to tackle shakes. I can use Tripod wherever possible but want to know how other methods being used by professional photographers. Which software is best in post-processing ?
In-lens VR is almost as good as in-body, if the lens system is good enough [they vary]. You ought to be able to see it switch on as you half-squeeze [it's on full-time in live view].
Nothing can really fix shake post-pro in stills, just use faster exposure times. Though some apps will have a go, it's rarely useful.
For longer focal lengths, such as 600mm at the long end of the lens you asked about in another question, properly implemented lens based stabilization is more effective than sensor based stabilization. That's assuming all else is equal, which it rarely ever is. But they're usually equal enough to maintain the principle. The reason this is the case is that the same amount of camera movement with a lens having a very narrow angle of view causes the blur to travel further across the frame than with a lens having a wider angle of view.
On the other hand, all else being equal, camera based stabilization works better with wider angle lenses. But the effects of the same amount of camera movement is not as great at wide angles as at narrower angles of view. Unless we're trying to shoot at insanely long exposure times while hand-holding it's not nearly as important to have any kind of stabilization with a wide angle lens as compared to a long focal length lens. If anything in the frame is in motion, the subject motion will be the limiting factor long before lack of IS will be at short focal lengths. Neither in body nor lens based stabilization do anything to tame subject motion. Only shorter exposure times can do that.
Shooting anything beyond about 300mm on a full frame (about 200mm on an APS-C body) requires improvements in the user's technique to get good results compared to shooting at shorter focal lengths. There's a real learning curve involved. It's probably even a bit steeper than the one when jumping from a fully automatic small sensor point-and-shoot type of camera or even a phone camera to your first larger sensor interchangeable lens camera.
We have a few existing questions here that address shooting with long focal length lenses:
View (in EVF) keeps moving after taking a shot with telephoto lens
Do my photos look blurrier on a newer, higher-resolution camera body because my lens has a problem?
Nikon D3300 - Trouble Focusing on Birds both Stationary & Moving
Sigma 150-600mm vs 300mm prime with extender?
We also have a few existing questions about processing blur out of an image after the fact. Essentially, it's a lot of effort for not much in return.
Is it possible to fix severe blur from camera shake after the fact?
How can slightly blurred photos be improved in post processing?
Can anyone recommend *freeware* to reduce motion blur by deconvolution?
What plugin for Photoshop can I use to remove camera-shake blur?
I can use Tripod wherever possible but want to know how other methods being used by professional photographers.
There aren't any widespread "other methods". Pros use the proper technique, including proper camera stabilization hardware, to reduce camera movement during exposure, not after. For pros time is money. Preventing camera movement is much easier and less time consuming than partially correcting it after the fact.