Can you ever travel through your opponent's squares?

I've heard that you can't but I would think that you could soar over one.

Can you L/C or hover (fly) through opponents squares as the powers/abilities ignore the effect of enemy bases on their movement?

Yes.

yes to l/c, hover/fly, phasing.

I'm pretty sure you can on hss..

l/c, hover/soar, phasing... these are the only ways to travel through a square that is occupied by a figure.

And if you'd notice... even if they rules didn't specifically say this... if you tried to travel in any other way... you'd have to stop and roll break away anyway.

the one actual problem with HSSpeed is that is gets tripped up by hindering terrain. They don't have to break away, but they can't travel over characters or hindering terrain without stopping.

HSS and grounded: no, you cannot travel "through" an opposing character.

HSS and flight: yes, you can travel "through" an opposing character.

that's right.. I forgot..