I like to think  of the core as 2 systems that work together to stabilize the body. The two systems are as follows:
The role of the outer unit is to generate movement and stabilize the spine during excursions of the upper and lower body. Additionally the outer unit allows for efficient full body movement by keeping the spine in proper alignment. Think of a volleyball player leaping and then spiking the ball. The core muscles generate rotation in the torso to increase the velocity on the spike and they maintain stability at the spine and prevent overextension, which allows the force from the jump and the torso rotation to transfer effectively to the shoulder and eventually to the ball.
Glutes (yes the lats and glutes are core muscles. They directly influence pelvic position, which directly influences lower back position. If we are going to agree that part of the role of the core is spinal stability then we have to include these muscles as part of the core)
Rectus Abdominus (the abz)
The inner unit is sort of like a box in the middle of your torso. The role of that box is to expand to create pressure within the abdomen. This pressure stabilizes the spine and allows for lifting of greater loads and transfer of force through the kinetic chain. The diaphragm in particular has a huge role in proper breathing mechanics and therefore spinal alignment.
Given what we now know about anatomy and function the natural next question is how best to train this area of the body for optimal performance, either in sporting activity, weightlifting or just daily life.
As previously mentioned one of the primary roles of the trunk musculature is to aid in transfer of force between upper and lower extremity. If every joint is an opportunity to lose force the spine is the area that has the greatest potential for loss. However, with a well trained inner unit and outer unit, force is transferred effectively from hips to shoulders or vice versa. In many people the inner unit is underactive and the outer unit over active from a stability standpoint and this can compromise force transfer and spinal stability, as well as lead to a higher likelihood of injury. This often occurs by overtraining the outer unit muscles (particularly the lats and rectus abdominus). These type of athletes typically appear to have strong core muscles but are challenged to perform even one repetition of a true stability task.
All show, No go
We can categorize our core training movements as follows
Let's look closely at each of the areas and see what sort of value they have and how to implement them in training.
Flexion - Spinal flexion exercises used to be all the rage in the fitness industry and the predominant method of training the abs; in the common gym going public they still are. Go to any gym and look on the mat area and you're likely to find a few people stretching and a group of people doing all sorts of crunches in pursuit of the elusive six pack. 
However, in recent years the crunch and situp have lost favor particularly due to the research of Dr. Stuart McGill who has repeatedly shown the stress that high repetition and/or loaded flexion can have on the lumbar spine. If you haven't read into any of his work I highly suggest that you do so. I agree that loaded flexion is a major no-no and that people who sit all day aren't doing themselves any favors either. This part is important so I will re-state it.
No loaded flexion and no high repetition flexion. That means tighten up your weightlifting technique and toss out that 1000 crunches per day plan.
With that said the spine should have the ability to bend and flex. We can train it through simple cat/camel positions to start and eventually progress to advanced tumbling techniques. These type of movements are actually a great dynamic warmup for people with no history of disk problems and a moderate level of coordination
One last note on the issue of spinal flexion: after a long restful night sleep when the spine has been horizontal for hours the hydration status of the disks is very high. This increased hydration status can cause higher likelihood of herniation or other disk injury. For this reason I avoid all tumbling and flexion based movements with my 6 and 7 am clients
Lateral Flexion -
Loaded lateral flexion has been shown to cause a lot of disk problems and this isn't a movement I recommend training with clients. If you want to include unloaded lateral flexion in your dynamic warmup I see no reason why you can't although I don't know that it would be the best use of your training time. Additionally this movement will hypertrophy the quadratus lumborum, a thick muscle running from your pelvis to your lumbar spine. In other words the muscle under your love handles. If that is an area that you want to make bigger while simultaneously doing long term damage to your disks then this exercise is for you!
There is a common misconception in the fitness community that a sore or painful lower back is caused by a weak lower back. In many cases that couldn't be further from the truth. Many low back pain sufferers are in their current predicament because they lack anterior core strength to keep the spine in a neutral position. This forces the lower back to work harder than necessary and put excessive extension based strain on the lumbar spine. Using a proper assessment protocol can help you see what the problem really is
Not a weak back, but maybe a painful one
Back extension is best trained isometrically and in conjunction with hip extension. Deadlift and squat variations are simple exercise staples that when performed for appropriate volume and with good technique provide sufficient back extension training. Someone with very specific training goals (like squatting 3x their body weight, may be well served to add in additional back extension training). As with many spinal movements, extension in an unloaded environment is also of significant value for restoring or maintaining proper mobility.
Rotation - If you haven't read about Mike Boyle's joint by joint theory I highly suggest you REVIEW IT. In it he discusses that the lumbar spine is intended to be a stable joint and the thoracic spine operates as a mobile joint, or series of joints in this case. With that in mind we can form a multiplanar approach to core training. Rotation exercises of the spine are excellent for developing range of motion, power and athletic ability, so long as the rotation is occurring through the thoracic spine and not in the presence of flexion.
Anti-flexion - These are the isometric back extension exercises I referenced earlier. Any exercise in which the load pulls you into a flexed (or round back) position and you must resist is an anti flexion exercise. Think squats, deadlifts, kettlebell swings, even some lunge variations
Anti-extension - This is the bulk of the "abs" training that I do with my own clients. The goal is to place the client in a position where resisting extension forces is the goal. Many clients will attempt to complete these movements by activating the hip flexors so keep an eye on anterior pelvic tilt to ensure proper technique. These type of exercises train the glutes and abdominal muscles to work in unison to allow force transfer across the spine. Some personal favorite exercises include barbell rollouts, Pushups, TRX fallouts and OH pressing
Anti-rotation - Developing rotation and power in multiple planes is great and is the foundation for success in a lot of sports (golf, tennis, lacrosse, hockey, etc) but an inability to control throughout a full range of motion can lead to wasted energy and poor performance. That is why anti-rotation drills are a staple in almost every sports program I design and a part of most general population programs as well. Usually I will focus the first couple of weeks of training on anti-rotation stability and training rotation range of motion and then transition to more powerful active rotation drills for the trunk
Anti-lateral flexion - Lateral flexion under load causes disk problems for a lot of people by putting excessive strain on the lateral components of the vertebrae. However, we can still strengthen those muscles isometrically and in a manner that has greater carryover to real life. Options here are endless. Anything in which you load one side of the body will cause the body to want to shift it's weight to easily accommodate the load. Single arm squats, single arms lunges, single arms presses etc.
So what would a daily routine look like for a client with the goal of "improved core strength" ? This will vary depending on a persons posture and injury history or the sport they participate in but lets keep this plan well rounded just to be thorough
Yoga flow and tumbling 10minutes (not first thing in the morning)
Deadlift (training anti-extension)
ss/ hip flexor mobility (not necessarily core training but restoring proper length to the hip flexor will keep the pelvis/lower back in good positions)
Overhead Press (training anti-extension)
TRX single arm row (anti-rotation)
ss/ pushup march (anti-extension and anti-rotation)
lateral lunge (not directly core training but anterior loading will cause some anterior core contribution)
ss/ pallof press (anti-rotation)
Sledgehammer swings (rotation)
farmers walk (anti-flexion and extension)
rowing ergometer (anti flexion)
A good full body routine with sufficient multiplanar challenge for the trunk all without doing a single crunch or side bend. This method of categorizing your core training will allow you to build a strong and stable core and keep a happy and healthy lower back. Now get to work!
 Obviously I am not the only person or the first person to categorize the core this way. If I could remember who I learned this concept from I would gladly give them credit. I want to say Paul Chek
 your six pack is not missing due to a lack of crunches, but rather a lack of disciplined eating and likely sleeping
 A vague goal but one that every personal trainer has heard at least a dozen times