Welcome to NavList.
"So on to my question. I've seen a few of the posts on this forum talk about calculators, and for completeness sake, I wanted to make sure I got as wide a range of opinions as possible. So here's my question: what calculators do you use for routine celestial navigation and why? Or do you use something else (a smart phone app, computer program, pen and paper only, etc), and why? "
My favorite method for doing cel-nav is paper and pen using Greg Rudzinski's " Ageton Classic Sight Reduction Table". Only 12 pages ( 6 back to back ) and you can do St Hilere intercepts, timed sights, great circle sailing, compass checks with amplutude or any other formula that uses sine and cosine. Fast and fun - no need for electronics.
I also have written several programs in Small Basic that I use when comparing a lot of ( other peoples ) sights. My latest project was teaching myself Excel and now use the spreadsheet to do sight reduction. But for fun I still like pen and paper.
My calculator is the TI-30Xa. Like Bill, I think it is fast and reliable. No solar power but after 10 years of using it the thing just keeps on working.
You asked why we use a method. For me, if I just want quick answers then my calculator or spreadsheet is the fastest but for fun, to go outside and take a sextant sight, come inside and calculate altitude and azimuth then the hands-on feeling of pen and paper, looking up data in my alamanc and sine tables is my favorite.
I am a retired pilot. When I flew and needed to know my position then speed was important but now, hands-on fun is my way.
Greg's Classic Ageton tables are avaible in the NavList arcives.