A "good go-to set of flats" would be Fender 9050Ls. I have them on a couple of my basses, and I've had them on a couple of others. I call them "Goldilocks" flats; not too dark; not too bright; not too stiff. They're...just right. With flats, unless you get outright bridge cables - like Roto SH77s or the "Jamerson" flats from La Bella - what you need to be concerned with is simple
stiffness. And, yes, pretty much all flats are pretty stiff. The Fenders, however, are not
real stiff. Not as flexible as TI Jazz Flats, but not as stiff as "normal" gauge La Bellas, either. If you want to try flats, the Fenders are an excellent place to start...