Raydiantdetail- Your "spin/twist" is kinda what I meant by the "scissoring" method. I won`t be doing that in this lifetime
Props to you again for your MA interest and passing it along to the next generation.
pgp- IMO situps aren`t *necessarily* bad for the back, but the back shouldn`t really be involved and when it is the poor form can lead to trouble. I don`t do those anyhow as they focus almost entirely on the Hip Flexors and the *upper* part of the abs, and those aren`t what need the most work IMO; I`m all about the lower abs and the obiques. Focusing on those still gives my upper-ab area plenty of work.
The big problem with situps is that they often work the hip flexors instead of the abs, and when those are overdeveloped (relative to abs) various things go haywire. So instead of situps, do crunches, like with your legs up on a bench/etc., rolling your shoulders up off the floor using just your abs (don`t pull on your head).
If your back *arches* when working abs, then something`s wrong, you don`t want that.
Eh, I`ll have to do some reviewing/studying before I can get very specific about Core Work as I`ve basically just done what works for me and only remembered info long enough to get squared away

But I will say that IMO most people go about it *all wrong*.
What I`m doing for my Core:
I exhale during the positive/concentric part of the motion, blowing air out between my teeth with my tongue pressed against the back of my upper teeth. Got that last part from Pavel...not sure if it really does the neuromuscular thing he says it does, but it`s kinda a habit now. But *do* exhale when doing each rep, don`t hold your breath or try to inhale at the wrong time.
I`m all about focusing on the Lower Abs and Obliques. The function of the lower abs is to raise the buttocks off the ground when lying on your back, so that`s the movement you want.
-Reverse Crunches. I do these lying on the floor, a flat bench, an inclined bench. I vary my leg position to alter the resistance. Idea is to lift the buttocks up and roll the hips towards your navel.
Easy starting point: Get in front of a wall, lie on your back about 12" away from it and press your lower back/lumbar region into the floor (NO ARCHING!) with your heels resting against the wall. Lift your buttocks up while thinking "roll hips up towards ribcage" so your heels come off the wall and your legs come back towards your head (as much as they will). Try to feel your lower abs working.
-Leg Raises. I do these in a "ab" or IIRC "Roman" chair, which makes them easier than when hanging from a pullup bar (less swaying). You can do them sitting on a bench too, but it doesn`t feel the same to me.
-The Ab Bench. This is a plate-loaded thing you sit on to do a type of situp. (Google it up and you oughta find it.) I dunno...I`d quit using it, hadn`t for years, felt it was too focused on the uppers (that again

). But I`ve put it back into rotation and that was probably good.
-Trunk Twists for Obliques. These *MUST BE DONE PROPERLY* or they`ll cause injury instead of the desired results! The resistance must come *from the side*, so *NO!* don`t put a barbell on your shoulders and twist! That`s the exact wrong thing! (Emphatic enough to get my point across?) I`m using a resistance band, but might use a cable setup if mine were adjustable for height.
Resistance from the side around torso level works the outer Obliques. I connect the band to something right around ribcage-height.
Resistance from a bit higher (I sit or kneel with the band at the same attachment point) works the inner obliques and I`m focusing on those.
The ROM for obliques is only about 90° so don`t overdo it by twisting too far.
Lying Side Crunch- Lie on your right side on the floor, right leg bent, left leg ~straight, right hand on left obliques (to feel them work), left hand a)over head = hard; b) around head = medium; c) extended towards feet = easiest. Use your left obliques to raise your right shoulder a little bit off the floor, thinking "up and forward a little" instead of "just up".
I DO NOT do sidebends with dumbbells. I`ll *very rarely* do them with a High Pulley Cable Machine (good for decompressing the spine, BTW) with minimal resistance.
NEVER do "scissor kicks" or other such stuff where you`re lying on the floor doing those movements. Very hard on the spine.
I`ve upped my rep range for all Core Work recently to where I`m doing lots of reps for many movements (only a few for the hard ones though :0 ). Once you`re lean enough, spot-reduction *DOES WORK* no matter what anybody (else) says and doing more reps, along with increasing the frequency to just about the point of overtraining them, is working great for me.
I`d do the "Reverse Cruch in front of the wall", some regular Cruches with your legs up on a bench/etc., and the Lying Side Crunch. See how those go and really try to feel the right parts of the right muscle groups working. If you feel your back, then something`s wrong with the form. Generally, think "rounded back, not arched".
Eh, the above is probably a jumbled mess but it`s the best I`m gonna do right now off the top of my head