I'm using a bright color of thread for better visibility; when doing this for real you'd want a thread that matches your sequins and/or fabric.
What you need:
Loose sequins - flat or cupped
Beading needle (most regular hand-sewing needles won't fit through seed beads)
Seed beads or tiny crystals (optional)
Thread conditioner or plain beeswax to keep your thread from tangling
Regular sewing thread or beading thread (your preference)
Step 1: Simple Overlap Stitch
This gives the same look that you get from pre-strung sequins.
Bring the needle up through the fabric and string one sequin. If you're using cupped sequins as shown, you can choose to have the cup facing up or down (it's personal preference and what effect you want). Some people find cup-side-down snags less.
Take the needle back down through the fabric just beside the sequin (stitching one side of the sequin down), and then come back up very close to the edge. String another sequin through the center and repeat. The overlap will cover your stitches and hide the thread.
Step 2: Backstitch
This is the same as a regular hand-sewing backstitch, just with sequins. It's good for when you need a super-flat line that won't snag. I find it works best on flat sequins, or cupped ones sewn cup side down. I'm right-handed, so I work right to left; reverse if you're left-handed.
Bring your needle up through the center of a sequin. Go back down into the fabric just to the side of the sequin, bringing it up a short distance on the other side. You want the needle to come out where the center of the next sequin will be, so evaluate that distance based on the size of your sequins.
Thread on a second sequin, then go back down through the center of the LAST sequin. Come back up on the far side of the second sequin and repeat - go back down through the last one, then forward.
Your thread WILL show with this stitch so you want a matching thread. For a fancier version, you can thread beads in-between each sequin to hide the thread; however this does give a more dimensional effect if you're going for super-flat.
Step 3: Sequins With Diagonal Beading
This is a great stitch for edging appliqué pieces or just outlining motifs.
Bring your needle up through the fabric and thread a sequin, then seed beads (the number of beads depends on the size of your sequins). Go back down through the fabric a short distance diagonally along the line you're outlining.
Come back up next to your first sequin; if you have to place the new sequin first to make sure your spacing is correct, do so. Repeat until you've got a nice line!
You can also do this backwards (beads first, then sequin to finish); it's a personal preference which way to work.
Step 4: Two-sequin Line With Beads
Bring the needle up and thread a sequin, then some seed beads (as before, the number depends on the size of your sequins). Thread another sequin (make sure they're facing the right way, if they're cupped sequins) and go back down through the fabric.
Bring the needle back up a short distance away; the distance is your preference. You can have the pairs right next to each other or have a bit more space.
Depending on how many beads you use and how tightly you pull them, pairs of cupped sequins will angle towards each other and create an interesting texture. If you use more beads that the distance requires, the beads will loop up in a small arch (again, gives a slightly different look).
Step 5: Individual Sequins With Bead Stopper
To sew single sequins without visible thread, bring the needle up through the center and then string one seed bead in a matching (or contrasting for a different effect) color.
Then go back down through the center of the sequin again, being careful not to go back through the bead a second time! This secures the sequin with the bead as a stopper. For extra flash you can use tiny crystals instead of seed beads.
You can also use this method to make short dangles or loops by threading multiple seed beads before going back through the sequin.
Step 6: Sequin Backstitch With Trailing Beads
Bring the needle up through the fabric and thread on a few beads (how many is your preference) and a sequin. Make sure all your sequins are facing the same way if they're cupped.
To make a line, go back down through the fabric and come back up to the left (to the right if you're left-handed and working left to right). The exact distance will be determined by how big your sequins are and how many beads you're using.
Thread on more beads and another sequin, and go back down right where your last stitch ended. Repeat.
To scatter this motif, take a few small stitches on the wrong side to secure everything after each bead/sequin combo is sewn down.