Your choice, first of all, will depend on medium you use.
For oil painting we’d recommend bristle brushes, because bristle hair spreads heavy paints easier and lighter. Bristle brushes work perfectly for applying paint, blending, some glazing techniques, heavy and rough brush strokes, as well as thin layering. All underpaintings on bare canvas or board should be done with bristle brushes. However, bristle brushes may over-blend colours turning them into thin dull leyer; especially if you work on wet canvas. To avoid this we’d recommend using synthetic brushes. They have softer hair and allow applying paints smoother and thicker with minimum of blending. Synthetic brushes work very well for glazing techniques and smaller details too.
For the most precious details, such as fine lines in portraits, animal hair or drapery, it may be better to use Kolinsky sable brushes. They are soft and flexible enough to hold oil paints, and at the same time they keep shapes better than synthetic brushes and apart from synthetic or bristle brushes release paint gradually. All these make painting details more controllable.
For acrylic painting we’d recommend bristle and synthetic brushes (see above).
Russian squirrel hair brushes are “must have” for watercolours. They are the softest brushes ever possible and work perfectly for both wet and dry techniques. Squirrel hair is able to hold a large amount of water and release it gradually. You can easily work on layering with this kind of brushes. Due to their softness squirrel brushes don’t blend layers, adding new ones on top instead. All these make painting in watercolours cleaner and colours purer.
Also Kolinsky sable brushes work perfectly for watercolours, but they are more expensive and have a little bit harder hair than squirrel ones. For cleaning and whitening techniques you should use synthetic brushes, because their hair is much harder and allows washing paints out easier without any damage to a brush.
For gouache we recommend synthetic brushes, because they are soft enough for working on layers and details and quite hard to be able to hold a large amount of paint. Natural hair is not a good choice for gouache.
These suggestions reflect our experience and a wide variety of techniques we use. Since no rules are set in arts, we don’t insist on anything particular.