Mobility work is very popular these days and rightly so, it is important regardless of the type of exercise you do.
Stability work on the other hand is definitely under used and to be quite honest it's because a lot of coaches/personal trainers haven't got a clue to what it really means.
Learning how to engage muscles will help stabilise the joints around them, especially the spine.
Let's start with the buzz word of the fitness industry....'The Core'.
It's not about just working your 'core' by doing crunches, toes to bars, planks and other weird stability ball movements that just put your body into awkward positions, its about learning how to engage the core muscles before using them.
All movements work the core to some degree but if you are not learning how to stabilise your core with engagement then you're not going to be getting much from those exercises.
The best way to really target your core muscles in my opinion is to do heavy carries, off-set loading and unilateral work.
An example of this is:
Farmers walks - These can be done with a trap bar, dumbbells and kettlebells, these will also improve your grip strength (which has been shown to have a direct link to general health).
Kettlebell single arm carry - The off-set load is great for making your core work super hard even if your core engagement isn't that good.
Single arm overhead carries - These require stability in your whole body in particular the shoulder joint and scapula.
Give your core a bit of love, learn how to engage muscles and stop faffing around with exercises that look hard but have no effectiveness to them.