Maintaining a certain level of flexibility is important. If you're not flexible at all, this is a problem for you in terms of your muscularity. You being really inflexible will end up causing you to sustain a bunch of muscle strains. So don't be the person who never stretches. Opposite that, don't be the person who has crazy flexibility goals either. If you become too flexible, you're going to end up with joint injuries and complications instead of muscle strains- and this is actually worse. So find your middle ground. Don't avoid stretching, and don't avoid time spent not stretching.
There's plenty of positive effects of stretching people constantly publish- and I haven't figured out why considering it doesn't seem to be a lucrative thing, but nonetheless, some of them are valid. Your stretching, in addition to limiting strains, can improve certain types of chronic pain, and sometimes give you some level of muscular adaptation on the hypertrophy end of things. This isn't comparable to lifting weights, but there is some level of passive tension resulting from stretching that can cause some addition of new sarcomeres and related contractile proteins (things that add size and strength).
The only benefit worth counting though is the flexibility change and how that impacts both mobility, risk for injury, and some types of chronic pain. As far as the chronic pain part is concerned, consider that somewhere in the range of 80% of all people eventually experience low back pain, and about 80% of that is a problem resulting from muscle. Tight hamstrings cause most of this. Your hammies connect to your pelvis. When they're tight, they're locking that pelvis down tight in the back. Bend over. That pelvis no longer has any give to it- it's locked into place and when you bend over, this causes huge stress to go into your low back. Does that make sense?
So your hamstrings are the single most important muscle to stretch. If you're going to stretch one muscle group, make it this. Everyone needs to stretch their hamstrings: golfers, bodybuilders, princesses, and so on. Every human being alive must stretch their hamstrings. If they don't, it's trouble. And if you decide to keep stretching after hamstrings, do the other muscles in and around your hips- basically your hip flexors and extensors. This will help keep further mobility and pain free torso range of motion. So stretch your hip flexors and glutes. Those are second most important. Third most important, I would say, would probably be chest. If your chest muscles get tight, it pulls your shoulders forward- rolling them inward. If this goes too far, you're going to have nerve problems in your arms. The brachial plexus is the series of nerves that innervate everything in your arms, hands, and some shoulderish stuff. If your shoulders rock too far forward, your clavicle and first rib be positioned in a way that can pinch down on your brachial plexus. You don't want nerve problems, so it doesn't hurt to stretch your chest. That's third most important, and actually fourth most important isn't really worth your time. So stick with those.
Stretching the hammies:
don't bend over. This doesn't stretch your hamstrings. It just irritates your low back. When you stand there and bend over to touch your toes, your center of gravity moves from navaly-region, to just in front of your body at about navalish-height. Being as your center of gravity is now outside your body, you have to flex your hamstrings to keep you from falling forward toward it. If you relax your hamstrings, you fall. If you don't relax your hamstrings, they aren't getting the effects of the stretch. So what is? Your low back. In a compromising way.
So instead, trouble-shoot a powerhouse. I'm kidding. That shouldn't make sense to you. Unless you're so absorbed with your Pilates that you think it does. What real people should do is elevate one foot on something in front of you, and lean into it, brining your head as close to your knee as you can. Then switch legs. A couple stretches on each leg and you'll be okay.
Stretching your hip flexors:
get down on one knee proposal style. Relax the back leg (the one with the knee on the ground), bring your body forward, and lean into something. For instance: you're on your left knee. Your right knee is up in front of you. Lean your right arm on your right knee, find something to lean your left arm on, and lean forward. You'll feel it in the hip flexors of your back leg. Then switch legs. One set of each will suffice.
Stretching your glutes:
lay down on your back, knees bent. Cross one ankle over the opposite knee. Bring the non-crossed leg up, wrap your hands around it, and pull it towards you. This will stretch your hip extensors and rotators, i.e. glutes, in your crossed leg. Switch legs. One or two sets on each leg and you're good.
Stretching your chest:
stand. Extend an arm out to your side, shoulder height, palm facing forward. Brace your hand up against something in that position, and gently ease your arm back further until you feel the stretch in your chest. If you keep your elbow bent, it may help your shoulder feel more pleasant during the stretch. Switch sides. One good stretch on each side is all that's really essential here. More is better, but one is pretty nice too.
Specifics of your stretches:
static holds. This is key. Bouncing doesn't do anything due to the fact that you have muscle spindles. Muscle spindles cause muscle contraction upon quick length changes. If you're bouncing during stretching, this is quick length change. When you do this, the muscle you're trying to stretch is contracting in opposition to your stretch. This gives you nothing on the results scale. So just be stable and hold it.
For the timing, make this static hold last for 20-30 seconds. Anything under that and you're not getting the full visoelastic potential. This being the permanent length change resulting from stretching, i.e. your goal. So nothing under 20 seconds. And then anything over 30 seconds isn't accomplishing much. You pretty much get all the results you're going to from a single stretch within the first 30 seconds- so anything over that is somewhat of a waste of time.
As far as when you should be stretching, make it after your workouts. Not before. People suggest that your risk of injury is lessened by stretching before. No. No it's not. However, if you stretch before, your max force production goes down (i.e. your strength) primarily because of the whole passive tension muscle spindle thing, but nerve activity can play in too. Point being you become temporarily less strong after stretching. And this is on top of the fact that it doesn't do you any good to stretch before. Not just on the injury-prevention side of things, but also, the visoelastic effects won't really happen. You want the muscles to have been used first. This warms them up and allows for a little more elasticity, so you can stretch them further, and then get the results of that. If you can't work-out first, contracting them on and off for a minute first or putting a heat-pad on them will help warm them up too. Of course if you ice the muscle, you slow down nerve conduction velocity, so you get less resistance and pain from the stretch on grounds of nerve activity. So technically an iced muscle can stretch pretty well too. But cold muscles strain easier, and heat tends to be more comfortable than ice, so sticking with heat is your best plan.
Regarding the frequency of your stretching, three to seven days per week. That's your range. So if you're looking to do the absolute minimum amount of stretching necessary to be a human being, it's this: two 20-30 second stretches on each hamstring, three days per week. Any more than that is unnecessarily good for you.