At long last...
... Google's Dataset search tool is now out of Beta and ready for your consideration.
You can use it to find open-access datasets. Just search on the topic of interest and you'll see a list of results on the left-side, and a summary of the dataset that's selected. For example, here I'm searching for datasets about snakes...
You can obviously scroll down the list on the left and see other datasets on this topic.
If you click through to the dataset you'll see the original public, with links to their data.
A couple of things to notice:
On the upper left of the page, you'll see a few filtering options (Updated date; Download format; Usage rights; Free). If you need a particular kind of data, use those to limit your results to what you need.
2. Autocomplete options
As you're entering your query, don't miss the popup autocompletions that will appear. (If you're looking at your hands, you'll totally miss this... yet another reason to learn touch-typing!)
Here's an example:
This is a great way to see what other dataset searchers are looking for (as a community). Perhaps it will give you some good ideas for your own search.
As it stands now, Dataset search indexes around 25 million datasets. Surely there's something of interest to you in there!
Here are a few examples I found that I like (you can imagine interesting uses for these):
a. Annotated database of cat images with facial features tagged
b. Dataset of MIDI tracks of musical performances (MAESTRO)
c. Crime and incarceration in the United States
d. Parrotfish density data. (You knew I'd search for this, yes?)
Explore, and enjoy!