Lightning design system provide static resources such as CSS and HTML there isn't any JavaScript. So, why slds-scrollable class is not compatible with s1 mobile?
Does this mean that I can't use my custom scroll CSS in s1 mobile?
I could not find resource about incompatible CSS properties. So, I don't think that this should be the case.
Best Answer
There's already a known issue on this, you can probably click "This Issue Affects me" button if you're interested.
You can use newly added Lightning component in the Winter '17, ui:scrollerWrapper that enables native scrolling in Lightning Components Salesforce1 app as well as in Lightning Desktop.
Please find the code below:
Component:-
Style:-
And, please find the following questions/issues regarding slds-scroll (since you've mentioned you didn't find any regarding that):-
There are several other questions and answers regarding the custom scroll functionality and the work arounds. (I've only listed a few of them above)
Anyway, you can definitely use ui:scrollerWrapper to enable scroll functionality in Salesforce1 app.
I understand this does not provide a direct answer to your question, but hope atleast the workaround of using
ui:scrollerWrapper
helps.