That's a tough question to answer because it depends on where you're going, who you work for/with, the purpose of your visit, the length of your visit, how you dress when not at work, what you do when not at work, etc...

My last two trips:
1) London, 10 days, visiting company office, casual environment: packed 2 pairs of running shoes, one pair of sneakers, wore boat shoes on the plane
2) Dallas, 8 days, visiting client site / client dinner: packed 1 pair of running shoes, one pair of dress shoes, one pair of flip flops and wore boat shoes on the plane