My weapon of choice is an EV PL88L, which is a lot like a Shure SM58, maybe with a little some bit of extra something character-wise. Maybe it's just the tan paint job, or maybe it really does have something special in its circuitry. Same as the 58, pop the filter off and it's a 57. I'll do that, then mix the microphone recording with the DI signal I run in parallel. Or with my setup, I might run clean DI, a Line Out DI from the head, and the mic'd cab signal. That's overkill but I like overkill
Basically, the lows aren't what give your bass the key characteristic tones, the mids are. You can capture lows aplenty with DI, and mixing it with the mic signal blends that full-range response with the midrange character your amp imparts. A standard-response mic will capture those mids in detail, and as far as I'm concerned, their tendency to roll down the lows makes mixing the signals MUCH easier than blending a bunch of super-low frequencies & subharmonics. Supercardioid mics like the Telefunken M80, EV RE20, Shure SM7B, etc are certainly usable and come with heightened proximity bass effect, but also cost a lot more as they excel in their designed purpose (and just coincidentally work well for bass recording).
Even if you get a crazy cheap mic, like a Pyle PDMICR 42SL, you'll have a mic signal and something to mix with DI. Cheap mics are fun because they can have unexpected characteristics, and as long as it's not noisy or anything, unexpected is worth exploring.
I have recorded at least one album track in the in the past with an SM-58. With the ball off it is pretty much a 57-equivalent.
El M is right re source. The amp doesn't have to be expensive, but it has to be DOING something you want (flooding, driving, harmonic enhancement ) to be better than a DI.
Instead of selling kits with both screw-on filters, Shure decided to sell the mics with different labels in order to sell more of 'em. Makes sense for profit purposes. But yeah, literally exactly the same microphone with different attachments; if you own an SM58, you also own an SM57 and vice versa.