(There is some good background info here: Pursuing the Perfect Embroidery Stitch-out.)

There are online tutorials that show how to use the water-soluble stuff called Badgemaster. I love the stuff for patches that can be washed.  Patches for Irish dance dress very often are made with fabrics that cannot be washed, so this is another approach.  The result is the same.

I use a LOT of this stuff -
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If you look online for tutorials involving this sticky stabilizer, you are told to cut your length of the stabilizer and then put it, paper back and all, into the outside hoop as below.
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Then you are told to score an area…
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…and pull off the paper.
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I dislike this because ultimately the hoop loses tension on the stabilizer (that paper is slippery!), and the fabric being embroidered can, and usually does, start to pucker and shift, especially when doing the complicated overs and unders of a celtic knot. I hate puckering and will do all I can to avoid it. I have yet to have a perfect embroidery sample, but I am working on it.

I pull the paper backing off the entire piece of sticky back stabilizer…
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…and attach it to my inner hoop.
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This next pic shows the sticky back attached tightly…when I thwack it with my finger it sounds like a drum. (Please excuse my “dirty” hoop…that is fabric dust, thread, sequins, etc, embedded in spray adhesive from my attempts to use the stuff years ago. It never goes away, but it is not bothering me or my fabrics!)
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Then I cut a piece of tear away stabilizer…
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…and then hoop the whole shebang.
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Next, your applique fabric must be interfaced and then stabilized with fusibles.  This interfacing and stabilizing helps the fabric stand up to what can be very dense stitching.  Whether or not your base fabric is stabilized with a fusible interfacing (tricot or woven) is totally up to you.  I personally feel that all fabric, including the heavy velvet, will support embroidery better if it is first interfaced with a fusible.  I do remove as much stabilizer as I can from the bodice and sleeves and soft skirts when the embroidery is done, but the interfacing remains to give the base fabric the body and support it needs both as clothing and as the base for the embroidery.

After the fabric is interfaced, I fuse Pellon Decorbond to the areas to be embroidered.  This is fairly stiff interfacing which helps support the stitches.  Yes, there is softer, cheaper stuff, but I have found it does not work as well as I want it to.  The excess Decorbond can be cut away, which is what I do, or it can be left to help achieve the level of stiffness desiredas when you make patches.

I will explain making this file into an applique patch:

applique panel bb by you.

Once you have hooped the sticky back and the tear-away, lay your interfaced and stabilized fabric in the hoop.  The following tackdown stitch will run:

tackdown by you.

Then the following knot will stitch out, stopping for the appropriate color changes:

knot by you.

When the above embroidery finishes, there will be another tackdown running stitch that runs just slightly outside the first running stitch.  I do this to help alleviate any fraying of the fabric. 

2nd tackdown by you.

Remove the hoop from your machine.  At this point, I always run a thin line of Fraycheck along the tackdown lines to stabilize the area further.  Next I cut very closely to the outside stitching as shown below:

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Then, put the hoop back on your machine and finishe the outside satin stitching:

applique panel bb by you.

Using other patches as an example, I will show you how I finish them.  Here, the patch stitching is done.
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Now I punch them out.
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As you can see, there is a fine fuzz surrounding the patches. This is from both the sticky and tear away stabilizers. This needs to be trimmed away. I either use my appliqué scissors or my fine, curved embroidery scissors.
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Here they are trimmed. I can get closer with the smaller scissors…obviously the trick here is to trim closely without cutting threads. Since the edge embroidery thread is white, this is where I stop. When it is colored, I use either a permanent Sharpie or a fabric dye pen in the right color, and I color in the outside edge so all the white is gone. This also further softens the stabilizer that is left which makes it lay flatter…you really cannot see it after that.
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I sometimes then iron a fusible web to the back of the applique to hold the patch securely in place when I the sew around the edge.  My preferred way of sewing around the patch is to use a clear monofilament, and using the outside satin stitching as a guide, I use a wide zig-zag to attach the patch flush to the fabric.