I have not personally looked into this, but surely there must be “apps” for various smart-phones that, in the various appropriate ways for each type of phone, implement “push notifications” and provide you the means to flip through them.   For instance, the ubiquitous “Facebook” app.   You get a notification that there is something out there that you need to look at, and then, with an appropriate app, you can retrieve and browse-through the list.

Neither e-mail nor SMS are (today) the best way to accomplish this task.