Totally depends on the year, how much, above where and the quality of it . However, you should figure on there being snow from the area of the fixed ropes onward. Most of the snow will be mixed rock, ice and snow. Don't wait too long to put on your crampons - there are not many good places to stop and put them on in the fixed ropes or afterwards. Look at the pictures posted on the Matterhorn page to get an idea of conditions encountered.
I went to the hörnligrat last sunday and there is still a lot of snow. even below the hörnlihütte. But if you go there on august, there won't be much left, only on the end of the route. usually soft snow. I recommend boots on which you can wear crampons for the last part of the ridge. And for descending, there are various routes, but on the normal route many people take the same way back passing by the solvayhütte
Usually one uses climbing boots with some flex and puts on the crampons at the Solvay hut. But this obviously depends on conditions a lot. Sometimes even the 'Dachl' near the summit has no snow at all.