I have been searching in the forum for a while looking for a solution for the store where a customer can search the variants -- in my case shoes sizes -- in a module and see the results on one page. Rather than having to open every product and see if the variant is in stock -- ie if the shoe model that caught his eye is in stock.
You should be able to do that with a filter applying on the characteristic. That would allow your users to select the shoe size they want and filter out the products without that variant or with a quantity of 0 for that variant.
You would also be able to create an input text filter in order to use your products listing as search results page instead of the joomla search.
I just tested that on our end and we fixed a bug on that whereby the quantity of variants wasn't taken into account. The fix will be in next version of HikaShop in the next few days.