Hip flexor stretches can be used to stretch the front of the hip joint.
The hip flexor muscles cross the front of the hip joint and when active can be used to close the front of the hip joint either by moving the front of the thigh towards the pelvis or by tilting the pelvis forwards, towards the thigh.
Note that the psoas major is a hip flexor and so these could also be thought of as psoas stretches.
In extended cat pose the focus is on lifting the leg to stretch the hip flexor.
Starting in cat pose, rather than keeping the spine straight, move the pubic bone back (between the thighs) and the tailbone forwards so that the pelvis tilts forwards. Let the lumbar spine (and thoracic spine bend backwards.)
The reason for tilting the pelvis forwards is to give the thigh room to move upwards at the hip joint.
Lift one leg, keep the knee pointing downwards. Pull upwards on the thigh.
Lift the leg as high as possible while keeping the knee straight.
So that it is possible to lift the leg high keeping tilting the pelvis forwards.
You could give the illusion of deepening the hip stretch by letting the chest sink at the shoulders (above right).
Work instead at increasing the forward tilt of the pelvis, increasing the back bend of the lumbar spine and from there using the hip extensors to lift the leg higher, so stretching the hip flexors.
To target the rectus femoris, which crosses both the hip joint and the knee joint, you can try bending the lifted knee and grabbing the foot. You can grab with the same side arm and work on balance or grab with the opposite arm. Try both since they give a slightly different hip flexor stretch.
In both of these variations try to pull the foot back. Pull the knee up (using the leg muscles) and pull the arm up (Using the arm muscles.) And continue to bend the spine backward (using the spinal erector muscles.)
The High Lunge hip flexor stretch has a similiar shape to extended cat pose except both feet are on the floor. The focus in this hip flexor stretch is on sinking the pelvis.
An initial stretch is to have one leg back with knee straight and the back of the foot on the floor or toes tucked under. The other leg is forwards with the knee bent and shin more or less vertical. With hands on the floor and elbows straight focus on sinking the pelvis down to stretch the hip flexors or the rearmost leg.
From there place both hands on the knee and lift the torso to increase the stretch. Continue to sink the pelvis downwards. As in cat pose, you can focus on drawing the back thigh upwards as you sink the pelvis downwards.
Press both hands down against the knee and use the shoulders to help reach the ribcage up and backwards.
From there reach the back leg arm up. Reach upwards through the ribs and arm, drawing the ribs up away from the pelvis.
Hold for a few breaths and then reach both hands up.
The ribs, arms and head all reach upwards while the pelvis sinks downwards.
The back thigh can be pulled upwards relative to the pelvis to increase the hip flexor stretch.
You could rest between each variation of the above hip stretch by straightening the front knee and bending forwards.
An option for less flexible practitioners is to use props as shown below.
A similiar style of hip flexor stretch is upright yoga pigeon hip stretch. In this pose the front shin is on the floor with the foot further back than the knee.
Read yoga pigeon pose for more on setting up the front knee and activating the front leg.
With the front knee bent, shin on the floor, position the back leg parallel to the long edge of your mat with the knee pointing downwards. Use your arms to push the ribcage upwards and backwards.
Use your spinal erectors to bend the lumbar and thoracic spine backwards. To make it easier to bend backwards try lifting your back ribs. Reach them upwards towards you head.
Using your arms and front leg push your pelvis back and down. Reach your back leg back.
You can experiment with lifting the back thigh relative to the pelvis. (The same action as used in extended cat pose and high lunge.)
As in high lunge, you can tuck the toes of the back foot under or have the top of the back foot flat on the floor.
I'd suggest using both options. Note that you may get a deeper hip flexor stretch with the toes tucked under since this lifts the knee higher when the knee is straight.
As in cat pose an option for targeting the rectus femoris is to bend the back knee and grab the foot. Do keep the spinal erectors active. And do continue to press down with the front knee.
Pull back with the lifted foot. You can grab it with the same side hand or the opposite hand.
One way to stretch the hip flexors of both legs at the same time is with upward facing dog.
Use the spine erectors to bend the lumbar and thoracic regions of the spine backwards. You can work at pushing the pelvis down while at the same time pulling upwards on the back of the thighs while keeping the knees straight.
A variation of this hip flexor stretch is to do it with the chest facing a wall. Place your hands on the wall and use the hands to push the ribcage backwards.
Keep the legs active, focus on reaching them back, also, make the neck feel long (pull your ears up away from your shoulders.) Play with your distance from the wall and work at gradually straightening the arms.
Other hip flexor stretches you can play with include half bow pose where from upward dog you grab one foot. Pull the foot back.
For bow pose, grab both ankles. Bend the spine backwards, pull the feet back, away from the hands, and pull up with the knees.
Place the hands on the floor behind you and lift the hips. Use both legs. Press down with the knee and press down with the foot at the same time.
Move the foot further forwards (or back) so that the knee is comfortable. Use the lifted knee leg to help stretch the hip flexor of the other leg.
Focus on moving your pelvis forwards as well as up. As it lifts higher you can move your hands forwards so that the hands stay under the shoulders.
From the previous pose, another possible hip stretch is to kneel with the hips lifted and push the hips forwards, either starting with the hands on the floor and keeping them there, or while lifted, dropping the hands down to the heels or the floor, and from there pushing the hips forwards.
If you are less flexible you may find it easier to start with your heels lifted (toes tucked under) and with your hips back a little. Lower your hands to your heels and then push your hips forwards to stretch your hip flexors.
Once you are comfortable with your hands on your heels you can try reaching your hands to the floor. You may notice that you can't push your hips quite as far forwards as a result.
Read more about camel yoga pose.
One fun way to stretch the hip flexors is while balancing on one leg in dancer. As a preparation for splits you can work at bending forwards and touching the free hand to the floor.
A very intense hip flexor stretch that actually starts of as a quad stretch is to position your self in a lunge with the back shin against a wall. To stretch the quad of the kneeling leg push your hips towards the wall. To target the hip flexors of that leg as well as the quadriceps, try to lean your upper back against the wall. Muscle control can help alot.
From here, move into a lunge. Move your hips forwards and rest your hands on the floor. If you like you could sink your chest to the floor in a low lunge to target the hip extensors of the front leg.
If you keep your torso upright, while at the same time moving your front foot further forwards and sinking your hips, then you'll be increasing the stretch of the back leg hip flexors.
Get low enough and you'll be doing a version of the splits. And for this reason I often like to use this hip flexor stretch as a preparation for splits.
One other Hip flexor stretch is front to back splits with the torso upright.
You can start with the front knee bent and the back knee on the floor. So that this stays a hip flexor stretch, keep the back knee pointing downwards.
As in high lunge and upright pigeon, try pulling up on the back of the back thigh. Pull forwards on the front of your lumbar spine but keep your torso upright. (The "pulling forwards" is to help activate the psoas muscle.) Use your arms to support the weight of your body, and as the hip flexors or your rear leg relax, let your pelvis sink deeper.
For combined hip flexor quad stretches, look at hero pose or virasana.
Here are more Hip stretches.
As a yoga teacher, I'm constantly exploring new exercises, new ways of doing yoga poses.
There is no single "right way" of doing a yoga pose. Instead, there are options. And the better you are at "feeling" your body, the better you can get at choosing the right option for your body as it is now.
For any technique, the point of practice is to learn feel it and to control it, so that it can be used without thinking about how to use it.
And that is more or less the approach taken in all of my ebooks and videos. They help you to feel your body and control it so that you can work towards using it effectively in anything that you do.