First of all, is a distortion pedal necessary for shred guitar? I’d say yes, insofar as your amp’s preamp section is lacking gobs of distortion and you don't possess the best technique. Why? Shredding is more or less synonymous with playing fast and, while it would be natural for you to assume that shredders, for the most part, possess above average technical chops, you’d be mistaken. The distortion pedal is useful for shredding because it will compress your sound and provide the illusion that you possess skills you do not have. Sometimes the distortion pedal has been referred to as "instant talent." There's truth to that.
You almost never see a shredder ‘shredding’ on a clean guitar or acoustic because they rely on distortion as a crutch. Distortion is just a word for higher order harmonic overtones, but in practice what you have is the more distortion generated, the more compression.
Distortion helps to level out all the notes you play (or misplay) so that you get an even sound — which falls apart as soon as the distortion box is switched off, revealing the fact that you are a hack with terrible technique.
There’s a baldheaded clown on youtube, for example, that teaches speed techniques for tweens who leans on the most distortion I’ve ever heard on a guitar. It might as well just be a synth patch. When you have that much distortion going you can just mush your way through a sequence and whether you hit the strings accurately or just scrape, ping, chop, or drop your pick altogether, your fretting hand can keep it going with nothing more than a shit ton of distortion flattening things out.
Which pedal is best for shredding? The one that emphasizes the most upper-midrange frequencies. You don’t want a lot of low end to muddy up affairs. An old Tube Screamer or one of the thousands of clones will get the job done — just crank it up and use the tone knob to make it bright. Also, some shredders will solo using a wah pedal as part of their sonic crutch because that is useful for carving out a lot of extraneous noise their bad technique engenders.
You’ll know you have the right distortion pedal when you find the one/setting that makes pinch harmonics the easiest. Go out a try a few and find the one that you can make squeal like a pig and you’re good to go.