Microsoft's position that "app store" is generic seems misguided. Marks are viewed along a spectrum of generic/descriptive/suggestive/arbitrary/fanciful, where generic marks are not capable of being protected, descriptive marks can become protected once they have acquired "secondary meaning" and suggestive, arbitrary and fanciful marks are capable of being protected, with varying strengths absent any requirement of secondary meaning. Microsoft is arguing that the mark is generic, and in the alternative, that if the mark is descriptive, that any secondary meaning is de facto and does not convert the descriptive mark to one being capable of trademark protection.
Genericity is something that markholders often spend a significant amount of time having to combat. By way of example, Xerox led a massive campaign encouraging people to "photocopy it" rather than to "Xerox it" because the term "Xerox" had begun to become ubiquitous with photocopying. Where a mark no longer indicates the source of the goods or services, but has rather become a categorical term of art to instead mean a type of good or service, it has probably become genericized. Markholders can combat this by actively protecting its mark against adoption by third parties as well as general usage by the public at large to mean the type of goods/services generally. In this instance, it appears that Apple has successfully combated genericism through actively contesting any third parties from using the term "app store" and instead imploring them to use app marketplace or application store. Notably, it further appears that Apple was the first provider to use the term "App store" and it's date of first use was March 6, 2008, and Apple filed its application on July 17, 2008. If the term has become commonly used, it was during the period that apple was actively using the mark and actively combating genericism.
The next issue is whether the mark would be descriptive. This one is trickier. I believe Apple would have a good "double entendre" argument here in so far as "App Store" could as easily mean the "Apple Store" and the "Application Store". A mark is descriptive essentially when the mark itself describes the goods or services sufficiently such that the consumer would know exactly what the goods or services are and is itself more of an identifier of the type of goods or services rather than the source of the goods or services. In contrast to a descriptive mark is a suggestive mark, which requires an inference or "mental leap" for a consumer to know what the goods or services are. In this case, "app" seems only descriptive insofar as it is the term coined and utilized by apple to reference its applications. As noted above, the element "App" could also mean "Apple" and consumers hearing the mark would then be required to make the mental leap to understand that the mark is the Apple Application Store. On the other hand, one point that supports Microsoft's position is that Apple refers to its mobile applications as "Apps" whether it is short for the term application, or in reference to the file extension .app. Notably, .app is not solely an Apple product. The use of "App Store" to indicate the source of "Apps" may then be descriptive. Nevertheless, Apple can then rely on the secondary meaning acquired by the mark. That would take extensive discovery, however I feel Apple would be in a pretty good position here. A quick Google search for app store shows pages and pages of Apple-only references, a good sign for Apple.