You will often hear people comment that they don't like machine quilting because it makes quilts stiff. The truth is it can and does. The good news is that it doesn't have to. The stiffness, is a result of the amount of quilting, the batting used, the thread used, the type of design quilted and the fabrics used.
If you want to use heavy threads, do but remember you need to quilt much further apart, at least a couple of inches and try not to cross the stitching too often. In a wall hanging this doesn't matter but on a bed or wrapped round a baby it does.
For dense quilting I use very fine threads most of the time.
I like YLI silk, top and bobbin, though to save money I sometime use Bottomline in the bobbin (it comes in loads of colours which almost disappear on the back of a quilt particularly if it is printed) If I am using White or Cream YlI silk, then almost any brand of white bobbin thread seems just as good. With this I can quilt as close as it like, often closer than 1/16th of an inch apart, without loosing drape qualities.
SewFine and Masterpiece from Superior in the top, and bobbin. These are nice threads, not as fine as the YLI silk but still fine. You can make them even more drapable by using an extra fine bobbin thread. With this I like to quilt to just to between 1/8th or a little less with no drape problems.
If I use the thicker threads I either use them for quilts which I consider utility quilts and wall hangings. If a quilt is going to be heavily used and not quilted exceptionally closely then these are perfect threads and my favourite of the heavier cottons is Aurifil. (I love the variegated ones). These threads are strong and will take the stress put on them by use if the quilting is far apart.
Battings make a difference, for the very best drape with heavy quilting, I love Hobbs Wool. When you have quilted heavily you will find you can wash your quilt happily in the machine as the close quilting stops the wool fibres rubbing together and felting and therefore shrinking. (Do make sure you use a wool wash detergent). Resists creasing.
Hobbs Cotton is good as it is not a thick bat, it drapes nearly as well as the wool. It can give your quilt an antique look once it has gone through the machine. And if it creases to remove the crease you need to either steam it or wash and dry.
Quilter's Dream Poly, is unlike other Polyester batts I have used. The polyester in this batt is microfibres and this mean the very fine fibres cling to your top just like the cotton. It comes in three weights and the finest drapes as well as wool. The middle weight is good for utility quilts and the heaviest is wonderful for faux trapunto work. It is warmer than cotton. I love it for quilts I want to wash frequently and tumble dry. The only downside I have found is that on using some cottons (but not all) it can beard (bearding is when fine fibres migrate through the quilt top). It is worth making a sample piece with the exact same fabrics as you have used in your top and backing, put it through the washing machine and dryer to see if the fabrics you have used are affected by bearding.
As I have mentioned before, the more a thread crosses another thread in your quilting motifs and even more in your background design, the more stiff your quilt will become. The one motif guaranteed to stiffen your work is tiny pebbles. Echo quilting does it the least.
I will assume you are using quilting cottons in your quilt top. These vary. If your top has loads of small pieces and/or has one or more of the more heavier coarser fabrics, this will make a difference. Some sheetings are so finely woven they can add to the problem. Look for a fabric with drape. This can be hard as often the fabric has so much finishing in it, that you cannot tell unless you wash it out first.
And remember if you want a quilt to be stiff and stay that way, do the opposite.
We have had a heavy frost here this morning and on the outside of the conservatory were these lovely ice patterns. These were taken from the inside.