If by “normal” you mean everyday shoes like you wear to the office then of course. Running shoes are engineered to keep you upright while moving at high speeds on a variety of surfaces both dry and, occasionally, wet. If, however, by “normal” you mean “normal” athletic shoes then the answer becomes more nuanced. Because what will give you more grip on the wooden floor of the basketball court is not going to provide very good grip when you’re careening down a mountain path and vice versa. The right way to approach the issue of grip when it comes to any type of athletic footwear is to consider the surface or surfaces you intend to use the shoes on. That will determine the type of sole your trail running shoes should have and the type of grip it should provide.