Being an avid skier I have ski poles lying around, so I've always used those.
Do proper trekking poles offer any advantages for snowshoeing?
If your shelter doesn't require them, I think the answer is "Ski poles are fine". Then, in the rare case where you want to walk without poles, you can tuck them horizontally in the small of your back (through your pack straps). If you like to go up or down with different length poles, just grasp the ski pole at a different spot on it's body.
Trekking poles are lighter and sometimes easier to break than ski poles. Some have the advantage of folding up into a short package.
If you use trekking poles, you may want some with changeable baskets so you can put on some larger baskets that won't sink into the powder so easily.
I have used both and it isn't really a difference to me. I would even prefer the ski poles, because they have baskets at the bottom, so that they can't sink deep into the snow. Trekking poles usually don't have this feature. So I would chose ski poles. But there are also two advantages if you're using trekking poles: Usually these poles you can adjust the length. This is very comfortable if you're wont to lend the poles, or if you cross a steep flank of a hill. In this case you can shorten one of the poles and increase the length of the other poles. Additionally, it is easier to carry them in your backpack because you can make them very short.
Definitively, I wouldn't use cross country ski poles, because they're usually just too long.
Yes, always use poles with baskets. Trekking poles allow adjustability which I find useful in changing snow conditions. ...but to snowshoe without poles? You can do it but it's so much more work and I cover much more distance with. JIMO.
In my experience, you really don't need either trekking poles or ski poles when snowshoeing. On flat ground or at low angles, I don't find them necessary or useful at all.
On very steep stuff, I find that an ice ax is more helpful (along with those mini-crampon things built into some snowshoes).
The only time when I've felt like poles would have been at all helpful was in certain very rare situations such as struggling up a steep gully into deep powder.
I've heard that you can get attachments to put on an ice ax that lengthen it somewhat and add a basket. This seems like it would be very handy, but I haven't been able to find anything like that for sale.