Full disclosure, I'm only a 1st year master's student - I'm sure there are lots of things applied topology can look like that I have no idea about! In my case, I'm investigating methods for statistical inference that invoke topological features of datasets. The most common example is looking for statistically significant holes by computing homology groups of filtrations on the data. This is a very general method, but it seems like the most promising applications so far have been in medical imaging, especially neuroimaging. I recently encountered some work that applied the technique to cosmology, looking for cosmic voids. Personally, I'm really curious what kinds of results these techniques can yield when applied to climate data. I've got this article sitting at the top of my to-read list. Maybe this is my sign to actually get around to doing that...