SQQ timing is taught by the Arthur Murray studios, while QQS is taught by Fred Astaire studios. The rhythm you learn will depend on which studio chain your teacher is a descendant of (directly or indirectly). Each of the two groups swears that theirs is the "correct" timing for Rumba.
Having taught for an Arthur Murrays for 5 years and then an ex-Fred Astaire studio for 5 years after that, I've had approximately equal exposure to both versions of Rumba timing. What I found was that I ended up liking the QQS version better in terms of the way the movements break down, and especially to commence the dance with a chasse. But purely from a timing standpoint, I don't see much sense in QQS. The bass, which is the backbone of the music, plays SQQ timing. So to dance QQS in my opinion is to go against the music.
The problem is, once you get accustomed to one or the other timing, it will start to sound correct to you, whether or not the music truly warrants it. This is why a Fred Astaire dancer will hear a Rumba as QQS, but will hear a Bolero as SQQ, even though the music can be similar in all respects except tempo. The orchestration can be different; A Rumba can be more rhythmical and a Bolero more lyrical, but none of these facts should offset the timing by two beats. This is why, after 5 years of each, I decided to go back to the original SQQ timing.
As for your Cha Cha timing, the ballroom dance world tends to be more unified: They like 2,3,4&1. There is a segment of the social dance population which prefers 1,2,3&4. What's interesting here is that the timing will tend to have an effect on the way you interpret the movement. Taking the rock step directly on the downbeat will have a more social (or even a cowboy cha cha) look, while taking the third cha on the downbeat will look and feel decidedly more ballroom (Latin ballroom, that is).
The timing of the music itself is slightly more flexible than Rumba. Cha Cha has a fairly straight and steady downbeat feel, with each of the 4 beats having roughly equal accent (apart from the implied strength of beat 1). Often there will be rhythms filling in between the beats, too (the "and" counts). Some songs may have a stronger accent on certain upbeats, while some do not. In traditional Afro Cuban-style Cha Chas, the Conga (and sometimes cowbell) plays a very noticeable "4&1" rhythm, which is why we try to match our "Cha Cha Cha" to beats 4&1 (and, as a result, the rock ends up on 2,3). But in other variations of Cha Cha music, an argument can definitely be made for placing the Cha Cha syncopated chasse somewhere else. There is a certain version of Tea for Two which has a strong 1,2,3&4 rhythm. Dance 2,3,4&1 ballroom timing to this tune, and you'll look like you're either deaf, or listening to a walkman.
Anyway, the moral of the story is that it's good to be flexible, especially where social dancing is concerned. If you dance with a partner who can't keep his or her rock step off of the 1 beat, then go with them. Social dancing is all about adaptating to conditions. That includes adapting to the styles of different partners, (different teachers in your case), and different types of music. Now that you know you can dance Rumba SQQ or QQS, you have more flexibility -- and that's a good thing!
On the other hand, if you ever dance for a ballroom exam or competition, make sure you have the correct timing down, even if the music is begging you to do otherise. Judges and examiners have a very fixed idea of what is correct in terms of timing: 2,3,4&1 for Cha Cha, SQQ for Rumba at Arthur Murrays, and QQS for Rumba at Fred Astaires.
Regards,
Jonathan Atkinson
For American style rumba, I'd say either SQQ or QQS is fine depending on what the music sounds like. I was taught one way (I forget because it was a while ago and I really only dance international now), but sometimes the music sounded one way to me and sometimes the other. I'd pick which ever rhythm seemed to fit the music better to me. As for cha cha, the standard is 2 3 4&1. Almost all ballroom cha cha songs are going to have a clear cha-cha-cha on the 4&1 count so the steps should match that music. I suppose you could dance it as 1 2 3&4 anyway, but since it doesn't match the music it probably wouldn't look right. I think Jonathan is right though, some songs don't put the cha-cha-cha on 4&1 in which case you should adjust your timing so that you match the music. I think Jonathan's example of Tea For Two is a good one; I had to think about it, but I do think he's right that at least some version puts the cha cha steps on 3&4 for that song. In that case, dancing the standard 2 3 4&1 wouldn't look right. I think the easy way to do it is listen for the cha cha cha in the music and then dance to match that, don't worry about whether you're breaking on 2 or 1 or whatever. Match the cha cha cha and you should be fine.