The best way to hide your thread ends is to avoid them whenever possible. This means that you should begin to "think" the pattern through before you tat it. Thinking ahead allows you to employ two shuttles, wind your shuttles in the continuous thread method (CMT), and/or to utilize the split ring, split chain, and mock picot to climb out of rounds. Eventually, no matter how hard you try to avoid it, you tat to the end of the thread, literally.
1. Don't bother to hide the ends at all if you are going to glue the tatting down or appliqué it to fabric. If gluing, then simply bring them to the back and glue. Later glue the entire piece down If appliquéing the lace to fabric, use a very fine crochet hook to pull the tail ends to the back side of the fabric and stitch down as you sew the lace on.
2. If the item will be seen on one side only and it is not planned to be worn, then the ends may be whip stitched to the back side of any ring or chain. Separate the threads in opposite directions and use a very fine sewing needle with either a sewing thread in a compatible color or use invisible thread.
3. If the item will be seen from both sides, but it is not planned to be worn, the thread ends may be "woven" back into the tatting:
a. Thread tails on fine blunt tapestry needle and weave under the "bars" of the double stitches around the ring or chain. Trim close and allow end to recede back into the tatting. Again, take each tail in opposite directions. This method is noticeable to the practiced eye.
b. Thread tail on fine blunt tapestry needle and insert into the bottom or end of closest ring or chain. Wiggle the needle so that it enters the "tunnel" formed by the carrying or foundation cord. If you have very tight tatting tension, this may not be possible for you, unless you plan ahead and tat the particular ring/chain to be used a little looser. Come up through the tatting about mid/ring/chain but not in the middle of a picot and trim close allowing the end to recede back into the tatting. This method may show a little if the ends of the tail are fuzzy.
c. If your thread is a soft multi strand twisted fiber, you can separate each tail into 2 or 3 strands, thread each end onto a sharp embroidery needle and "sew" into the ring /chain. Be aware that separating the fibers destroys the tensile strength of the thread and it will disintegrate when stressed. This method seldom shows.
However, tatting over the tails and encapsulation are the preferred method to hide ends. Please see "Hiding Tails preferred method" for additional info.