These are comparisons of the Arthur Murray dance syllabus, DVIDA, and US ISTD. All are for the American style dances and all are limited to the bronze level (including pre-bronze). I have not formally studied ISTD though I have reviewed for the sake of comparison. I'll be looking at each dance and saying which syllabus I favor and for what reasons, then giving overall conclusions.
Waltz - Major props to Arthur Murray here for a very innovative collection of steps that not only have exceptional variety, but do a great job of layering technique while also providing clear movement across the floor. ISTD gets the movement aspect right but has very little variety, including leaving out some fairly common figures for social dancing. DVIDA has decent variety, though not as much as I'd like, but doesn't provide nearly enough movement until the latter half of the syllabus; on the other hand, theirs is the best set up to learn the technique.
Tango - DVIDA's tango is interesting but thrusts people into tricky moves far too early and is awful at teaching promenade. Moreover, it changes alignment constantly and requires great accuracy of orientation, making navigation rather difficult. Murray's is just weird, rather dull in the first half filled with strange variations, then random movements in the second half. ISTD, on the other hand, focuses on core, useable actions, and arranges patterns in a very natural form of development.
Foxtrot - DVIDA loses this one right off the bat. Far too much of their bronze foxtrot syllabus tries to awkwardly shoe-horn in SSQQ timing, their awkward versions of feather and three-step are perplexing at best, and what the Hell did they do to the grapevine? Arthur Murray actually does a fantastic job, including developing alternate timings, natural uses of dance positions, and interesting variations to keep everything fresh. ISTD starts off well enough with excellent progressive steps, but the full-bronze portion simply feels like a rehash of waltz.
Winner: Arthur Murray.
Viennese Waltz - This is a hard one to evaluate as there are so many different approaches to American Viennese and so much disagreement about what the dance is supposed to look like at this level. DVIDA takes the approach of slowly developing the speed and drive the dance requires, then teaching the core Viennese movements, and only then developing the uniquely American patterns. Arthur Murray takes the opposite approach, beginning with these strange patterns almost right away. ISTD takes the worst of both worlds, teaching the core Viennese patterns, then the unique American patterns to develop the Viennese technique, and then to the difficult American patterns. On the other hand, ISTD has the best technique descriptions of any of the syllabi. It's a toss up, but gradual development wins out.
Smooth Overall - Each syllabus won at least one dance, though DVIDA narrowly won a second, but they suffer from their overwhelming defeat in foxtrot and often poor descriptions of technique. Arthur Murray has excellent patterns but the technique is often lacking from the syllabus, relying on a franchisee to impart on the teacher how the dance is supposed to work. ISTD has the best technique but tends to lack for interesting steps.
Winner: Three-way draw.
Cha Cha - ISTD begins with a very traditional, almost conservative, selection of steps. DVIDA's is only slightly more varied. Arthur Murray, despite a few unique variations, doesn't add that much to the dance's repertoire. The decision must be made by technique. Due to its variety of open actions, DVIDA just slightly edges out here due to its ability to use (and acknowledgment of the legitimacy of) straight leg actions.
Rumba - Arthur Murray bronze rumba? Dull as Hell early on, then confusing in the latter half. ISTD gets points for expanding a little bit and having great technique descriptions, but not enough. While DVIDA starts out shakey, with a strange side-basic and overuse of side-close actions (rather than 5th position endings), the full syllabus includes some amazing patterns that still fit the bronze mold - and even those side-close endings start to grow on you.
East Coast Swing - DVIDA is a bit strange here, leaving out several key steps (outside turns, lindy whips, points and kicks), and while the result is danceable, it might be very hard for a follower to dance with a partner with different training. It does, however, give great description of technique and feels quite jive-like. ISTD has a good mix of moves and good technique as always, but the controversial start on a triple makes some steps problematic. Arthur Murray, on the other hand, uses a good mix of steps with solid technique that result in a fantastic swing for social or competitive dancing.
Winner: Arthur Murray
Bolero - The ISTD syllabus feels largely like a rumba knock-off, and while one can uses those steps to learn the technique without learning new figures, it still doesn't feel like bolero. Arthur Murray's bolero is rather convoluted and becomes awkward; furthermore, requiring dancers to wait until full-bronze before even starting the dance can be infuriating. DVIDA has an amazing selection of steps, though the written versions often use dumbed-down technique; still, it's a fun and fascinating syllabus to dissect.
Mambo - Props to DVIDA for a fun collection of steps but they lose said props due to very strange endings on many figures (though the hip lift is fun). Arthur Murray does a good job with timing and technique, but ultimately their mambo just lacks variety. While ISTD starts off slow, the second half of the syllabus is simply fun.
Rhythm Overall - DVIDA clearly dominates the rhythm syllabi and I can't make an argument why they shouldn't. Even when they weren't number one, they usually had a solid case.