Why are all towel bars 24 inch? With studs being 16 inches apart - how do people mount them?
Drywall Anchors

I have had 100% success with drywall anchors but I am very careful about all aspects of the mounting hardware.  You have to design and execute your plan carefully to have success, otherwise they'll be vulnerable to pulling out as Michael Karas points out.  If you mess up any aspect of what I set out below, your anchors will probably pull out.  If, on the other hand, you get it right, the anchors will be stronger than the sheetrock.

Select a Towel Bar with Appropriate Mounting Plates

Select a towel bar where the mounting plate at each end has the following attributes:


holes for at least 4 fasteners - You can only expect about 50 lb pullout strength from each drywall anchor, but the pullout strengths add up.  So with 8 anchors altogether you end up with 400 lb pullout strength.
holes for 2 of the fasteners separated vertically by at least 1-1/4" - A towel bar is an overhung load so the tendency is for the top of the mounting plate to pull away from the drywall.  Fasteners near the top and bottom will prevent gaps from forming between the mounting plate and the drywall when the towel rack is under load.  You want a tight fit between the mounting plate and drywall because you need to at all times maximize the surface area in contact between the mounting plate and drywall.  Drywall fails when it is subject to pressure (ie. force per unit area) above a certain, rather small threshold.  Keeping the full area of the mounting plate in contact with the drywall at all times spreads the load over the widest possible area.  Area is the denominator for pressure (i.e. P=F/A) so maximum area minimizes the pressure.
solid flat-back mounting plate that encompasses at least a 2" round circular area - A flat back and large area minimizes the pressure on any single point on the drywall which is good for the reasons in the previous bullet.


This is an example of a good mounting plate (installation guide):

  

Note the flat, solid mounting plate and four-hole bolt pattern.

Select Studs or Pristine Areas of Sheetrock for Mount Points

Areas of sheetrock that are in any way damaged (from installation or otherwise) will reduce the pullout strength of the anchors dramatically.  I use a power drill with a 1/16" drill bit to test how sound an area of sheetrock is.  With a little practice you should be able to detect the difference between seams, drywall mud, studs, and pristine sheet rock.  In general you want to avoid the following areas for mount points:


anywhere within 4" of any edge of a piece of sheet rock including seams and corners
anywhere within 2" of a stud - There's a good chance there are drywall screws near that spot on the stud and drywall screws damage the sheetrock in about a 1" radius around the screw head.  On the other hand, if you can put the fasteners right into the stud, that is likely a better solution than anchors for that end of the towel bar.
anywhere there isn't enough clearance for the anchor and screw behind the drywall - The risk here is that the end of the anchor or screw contact a hard object (like an electrical box or concrete) and the anchor or screw acts as a jack against that object.  That will certainly cause that anchor to fail because you will immediately exceed the holding strength of the anchor when the jacking starts.


Select Quality Deep Thread Screw-Type Anchors

Forget about toggle anchors: The toggles are so bulky and the bolt patterns so crowded that the toggles will usually interfere with one another and whatever happens to be behind the drywall.

Sheetrock only has strength because the gypsum is under compression.  The gypsum is under compression only insofar the paper (yes the outside of drywall is paper) retains its tensile strength and gypsum does not leak out.  These anchors, used correctly, work without compromising these critical aspects of the sheetrock:



Note how the anchor's threads achieve a large contact area with the gypsum and have deep threads.  Note also how the head and barrel of the anchor seal against the paper to keep the gypsum in.  

When these anchors are installed correctly they do two critical things to preserve the integrity of the sheetrock:


They stretch the paper around the barrel preserving the paper's tensile strength.
Their barrels displace the gypsum slightly into the surround area thereby preserving or slightly increasing the compression of the gypsum.


If your method of installing the anchors doesn't achieve these critical things, you need to re-think your approach because the anchors won't hold.

For reference, here is the data on the anchors I use (I got them from Home Depot):

 

Note the table of shear and pullout strengths -- these are quality engineered anchors.  Installed correctly, they have real usable anchor strength.

You have to get your layout perfect, the first time.

You need to get your layout perfect the first time.  You cannot undo a hole, or move an anchor 1/4" to the right or left.  Once you put an anchor in the drywall you must leave it in there if there are other anchors anywhere nearby.  Otherwise the sheetrock in that area will be forever weakened and your other anchors' pullout strength will be severely weakened.  You can easily sink an anchor into the paper and float mud and paint over it.  You can't however move the location of an anchor once you have drilled for it.  Here is my advice for layout:


  Measure, check, measure, dryfit, measure, check for level, check what's behind the drywall, go for coffee, check it again, dryfit again, check for level again, and then repeat.  You must be absolutely sure your layout is correct before you drill your first hole.


Pilot Drill at 75% the Barrel Diameter

Forget what it says on the box of anchors about self-drilling.  You must pilot drill every anchor hole.  No exceptions.  Without a pilot hole, the anchor tips will do two things that are terrible:


tear the paper - This undermines the tensile strength of the sheetrock in the surround area.  If you do this, you no longer have sheetrock in that area. You just have a bunch of paper and dust.
wander - It is critical that your anchors line up with the hole pattern of the mounting plates (which you carefully layed out, right?).  If the anchor and hole patterns don't line up well enough, the anchors and mounting plate will fight one another, the shear strength of one or more of the anchors will be exceeded and those anchors will no longer have much holding strength.  Without a pilot hole, the anchors will wander before they start to penetrate, and you are basically guaranteeing anchor failure because the anchor pattern won't match the hole pattern.


The pilot drill should be sized at about 75% the barrel diameter.  This will result in the right amount of gypsum displacement and will stretch the paper around the barrel for a tight fit without tearing it.

Set the Anchor with a Slow Drill set to a Low-Torque Clutch

This point is extremely important.  If you overtighten the anchor into the sheetrock, the anchor will fail.  If the axis of the anchor changes as it is screwed into the sheetrock, the anchor will fail.  The only reliable method I have found for setting the anchors into sheetrock is with a decent variable-speed cordless drill starting with the following settings:


low gear
lowest torque clutch setting


Do not use an impact driver.

The reasons I use a drill are as follows:


you reliably can apply torque and stay on axis - As you screw the anchor into the dry wall it is critical to keep the axis of the anchor perfectly still.  Until the anchor is set all the way into the sheetrock, the gypsum is surprisingly vulnerable to damage.  Even the slightest change in angle caused by the driving tool while the anchor is being screwed into the wall will cause the gypsum around the anchor threads to turn from its semi-solid form into a powder.  When that happens, the strength of the drywall anchor is reduced.
you can drive rather quickly which reduces risk of damage - Once you have the clutch set to the right setting, you can set an anchor in one or two seconds.  This speed is an advantage because while the anchor is screwed part-way into the sheetrock, it is very vulnerable to damaging the surrounding sheetrock and undermining its strength once set.  With the speed of a properly-setup drill, you reduce this risk simply by getting through the vulnerable stage quicker.  You only have to hold the drill perfectly on-axis for seconds (see previous bullet).


Start twisting the anchor into the pilot hole with the drill.  Make sure you keep the axis of the drill completely still.  If the clutch is slipping, increase the torque setting just enough that the anchor advances into the wall.  Stop as soon as the head of the anchor makes contact with the paper.  The clutch should slip just as the anchor this occurs.  You might have to adjust the clutch to get this right.  The head and paper should be snug when you are done.  Do not overtighten.  Do not back out.  Doing either will almost certainly result in tearing the paper or losing some gypsum which will cause your anchor to fail.

Use the screws that came with the anchors.

The anchors and screws that go into the anchors are engineered to work together.  Don't try to substitute.  Their length and thread profile must be correct for the anchors to work correctly.

Assemble the towel bar to the mounting plates.

Once the mounting plates are installed in the correct position, the rest is just a matter of following the manufacturer's directions to assemble the towel bar.

Here is what one of my installations look like:

 

Note that even though the towel bar is overhung about 6", the mounting is plenty strong enough.  People knock their elbows on it and grab it to steady themselves without a problem.  This works because the load is distributed across 8 correctly-installed anchors for a combined pull-out strength of about 640 lbs and a shear strength of about 960 lbs.  At that strength, the weakness is the sheetrock itself, not the anchors.  In other words, to break this mounting you'd have to apply enough force to tear a whole section of sheetrock off the wall.