I'd like to see something like the 90's effect. It would make sense if Spider-man actually experiences some kind of visual distortion when the spider-sense is triggered. Like, maybe when the tingling starts his other senses distort in weird ways, creating dream like, hallucinatory effects. Cerebral effects, like fading, echoing and visual static. Colours can change and the environment shifts and breaths like waves are going through it.
The 90's cartoon would fade the background into black with strange abstract patterns everywhere, it was almost like for a moment he was seeing a mysterious alternate reality. That's cool to me.
I don't really see the spider-sense as precognition, it can appear that way because it allows him to predict incoming attacks, but it's not like you can point a gun at his head and make him guess lottery numbers.
I see it more like a 360 degree radar that is generated by 'feeling' things in the environment that are outside the range of human perception. Maybe it feels vibrations, heat, electromagnetic waves etc and it creates a tingling sensation that allows spider-man to feel the presence of things over a distance. But for what ever reason, it only activates in the presence of a threat, or during deep concentration.