Is there an easily visible consequence of special relativity that you can see without specialized equipment?

A working GPS receiver.

In general, things like a smartphone "verify" a great deal of modern science.

Yah. Though the immediacy of the verification will vary. When I use my cell phone, I really feel it that information is being carried by radio waves that don't penetrate metal. But I never found the GPS example quite compelling; people assure me "oh yes we needed relativity to get it to work right" and of course I believe them, but I've never seen the details presented and so this doesn't impress me at an emotional level.

I don't know how much my feelings here are idiosyncratic; how similar are different people in what sorts of observations make a big impression on them?

Just direct observation, by the way, gives you little. Yes, you can observe discontinuous spectra of fluorescent lights. So what? This does not prove quantum mechanics in any way, this is merely consistent with quantum mechanics, just as it is consistent with a large variety of other explanations.

I'm not so sure about "consistent with a large variety of other explanations" -- my impression is that nobody was able to come up with a believable theory of spectroscopy before Bohr. Can you point to a non-quantum explanation that ever seemed plausible? Furthermore once you say "okay, spectral lines are due to electron energy-level transitions", you wind up intellectually committed to a whole lot of other things, notably the Pauli exclusion rule.