That said, I think a lot of engineers out there have been burned (sometimes literally) by using tantalum caps in the past and are likely not to design them in in the first place these days. I know we moved away from them completely in my last job because enough exploded to scare people off.
EDIT: See dragontamer's comment below for more. Explained better than I did.
I briefly looked at the datasheet for the regulator and it has some constraints for the output cap that should help guide your choices. For example, you need to look at what temperature range you expect the regulator/cap combo to be working at, as certain dielectrics are sensitive to temperature changes [1]. If you need stability over a wide range, you might go with an X7R or better, otherwise X5R might do. You will also have to look at derating the capacitance for the operating temperature - you might find that in a hot environment you actually need a 47uF cap or greater to meet the minimum 22uF output capacitance requirement of the regulator.
The other main thing to consider is operating voltage. The regulator you list is adjustable, so you will need to pick an output cap that has an appropriate capacitance at the voltage you plan to run the regulator at. The curves drop pretty steeply, so it's really important to analyze this.
For ESR you should look at manufacturer websites. They often have a "picker" tool that shows graphs of that (it's dependent on frequency!). You probably won't have trouble finding caps with a low enough ESR value for this application as long as you stay away from electrolytics.
Also, you can consider paralleling some lower value capacitors to achieve the capacitance you need, and reduce the ESR a little. I wouldn't use it as an ESR lowering strategy however [2]
[1]: https://www.maximintegrated.com/en/app-notes/index.mvp/id/55...
[2]: http://electronics.stackexchange.com/questions/152549/esr-an...
Multiple scarcity events, skyrocketing prices, and then in 2014 you couldn't buy them even if you wanted to.
Purchasing agents were seen chasing design engineers with pitchforks by torchlight.
Ceramic MLCC's have filled their spot. Better, much easier to use and safer.
And the modern poly hybrid electrolytics are amazing, as well.
Tantalum poly caps are probably great, but the fear is still there and too recent - so I doubt they will get very popular except in small specific instances where they are absolutely better than anything else.