I am wondering what Ski Resorts are available (open) for skiing during the Northern Hemisphere's summer?
Level of comfort and amenities don't really make a difference for the purposes of this question.
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In Switzerland, there are 2 ski resorts where it is possible to go skiing in summer. This is because of the glaciers there. The ski resorts are:
Saas-Fee: Saas-Fee is not open year-round, but the summer season is between: 14.07.2012- 28.10.2012. For 2013 you can assume that more or less the same dates are available.
Zermatt: Zermatt is open year-round. You can go skiing there every day. Obviously, during the summer, not all lifts are open. But nevertheless, there are halfpipes, jumps, etc. during the summer on the glacier.
Although the other answers are OK, I will use the opportunity to advertise one peculiarity in the area where I live: midnight sun skiing..
Yes, you read that right. The ski resort of Riksgränsen, Sweden advertises with midnight daylight skiing, possible every year in May, and midnight sun skiing. Midnight Sun skiing is usually heli-skiing offered a few days around midsummer, and not guaranteed to be possible.
The ski resort is small compared to resorts in the Alps, but the possibility to ski 24 hours per day(*) should be quite unique.
(*) I don't think the lifts are open 24 hours per day.
I know of a few options you can have in the Northern Hemisphere that may appeal:
Timeberline Lodge in Oregon, USA - open twelve months a year.
Whistler-Blackcomb, British Columbia, Canada. I was there 3 days ago :) Has a glacier you can ski on in summer!
Italy has quite a bit of summer skiing at Cervinia Zermatt, Italy. The link is some photos showing you what's available.
More unusual options
Indoor skiing - there are several of these, the most famous being Ski Dubai. But they also exist in India, Belgium, China, France, Germany, Japan, Lithuania, Netherlands, Poland, Serbia, South Korea, Spain, and Russia (and places in the southern hemisphere too)
Heli-skiing. It's harder and a bit more 'extreme', but you can get to the snow that isn't normally available to you from a marked trail or lift.
Grass Skiing. Cough. Yes it's actually a thing, and no...there's no snow. But it's growing in popularity in both North America and Europe.
Summer skiing is definitely a thing. For Europe, I found an article that says many places used to offer it, but now it's pretty much just Les Deux Alpes, near Grenoble. But then this article disagrees and lists a bunch of others (and has pictures.)
And according to this Globe and Mail article, if you head to Whistler you can be taken by bus up to a glacier where you can ski even in the summer. They sell combo ski-ing and mountain-biking day passes :-)
I even found a Top Ten list that claims there is only one location in the US. It cheats by using Southern Hemisphere locations to offer you local-winter skiing during your home-summer, which I suspect you didn't want. So your list of possible locations dwindles to a handful.
Norway also has summer skiing spots.
On the tourism website for Norway, some of them are listed, claiming they're the largest :
the Stryn Summer Ski Centre, at elevation 1600m
the Galdhøpiggen Summer Ski Centre, at elevation 2600m, claiming some powder snow.
the Folgefonna Summer Ski Centre, which is a glacier.
I am surprised about the northern hemisphere bias, especially since @karlson explicilty states not beong bound to the northern hemisphere. I would say go to Chile. Some years ago, while traveling through Chile I went skiing just an hour drive from Santiago. It was a resort with modern facilities. I didn't even have to stay there, since day trips from Santiago were feasible. In the bus to the resort, I spoke to someone who told me that there were several other resorts spread over the country.