What are the differences between female Modern Orthodox/Modern Orthodox Liberal Jews and Conservatives in terms of women's modesty? My question applies both to dress and social interaction - shaking hands with men, singing or performing in public (not at the shul), wearing pants outside of the shul, married women covering their hair, mixing in public places such as work or swimming pools, etc. I would like to know the 'official' stance of the various groups and how it is in practice.
There are vast differences just among the modern Orthodox. If you want "official", you'd have to define some authority to whom everyone subscribes, which is impossible.
The United Synagogue of Conservative Judaism issues rulings that are supposed to be binding, but aren't, on all member synagogues. Much less are they followed by individuals.
Orthodox Judaism doesn't have any such centrality, save in a few places in which the government has sanctioned an official chief rabbinate, like the UK or Israel, for example.
Orthodoxy, however, by definition, lends itself to following authority, but it's a question of whose.
Many modern Orthodox Jews follow the same standards as more "ultra-orthodox" Jews on virtually all matters of Halachah, and merely engage more in modern life, so that their standards of modesty are totally within the parameters set down in writing in previous generations. Others, though, see modesty as subjective to the standards of society at large, and therefore, while still adhering to a standard (which may be hard for other groups to identify), is much more in line with what one might see elsewhere in society.
I couldn't find any relevant teshuvot on the Rabbinical Assembly web site, and as already noted by Seth J, there isn't a single authority for Orthodox (Modern or otherwise). I can describe what I have observed (primarily in one city). I often attend a Conservative shul for weekday services and occasionally others, and I've visited several Orthodox congregations a few times.
Women in Conservative synagogues may wear skirts, dresses, or pants. Sleeves may be short or long, but necklines are moderate to high. Covering hair appears to be uncommon. Women and men shake hands and there is mixed seating at kiddushes. Social activities do not appear to be gender-segregated. Some women work outside the home and do not limit themselves to gender-segregated jobs. I don't know what happens at public swimming pools.
It's harder to tell which synagogues are "Modern Orthodox" versus some other Orthodox designation. I've been to Orthodox synagogues where men freely shook women's hands and where women wore short sleeves (not pants, though). I've been to others where these things don't happen. This is somewhat confusing. A woman who told me she is Modern Orthodox dresses the way I described Conservative above (but no pants in shul), and she has both male and female coworkers.
My experience (of the American scene) is that there are numerous shades of Jewish observance, but that there is a particularly significant split between Conservative and Orthodox denominations with regards to women's modesty and separation of the sexes.
I think it is appropriate to draw up three (mostly) distinct categories:
Modern Orthodox (mainstream) / M.O. (Liberal) / Conservative
Shaking hands with men/women: In my experience, all Conservatives and most M.O. (liberal), as well as a a lot of M.O. (mainstream), have no problem with it. However, as you get to the right-wing side of mainstream Modern Orthodoxy, people start being more strict about this. Often, a handshake for reasons of politeness is still appropriate, but there is hesitancy with more gratuitously affectionate forms of touch between men and women.
Women's singing/performing in public (not in shul): Again, this goes from being very much not a problem (for Conservatives) to not a problem (for M.O. liberals) to often/sometimes not a problem (for mainstream M.O. people). It becomes more of a problem as you get to the center point of M.O. and rightward. Even so, a lot of Modern Orthodox people do not worry about the kol isha prohibition (prohibition on hearing women's singing voices) as much as those in non-M.O. strains of Orthodoxy.
Pants outside shul - You will by hard-pressed to find any Conservatives who have a problem with pants in or out of shul. Liberal Modern Orthodoxers may go either way on this; a lot of the women in this category wear skirts most of the time. Once you get to mainstream M.O. and rightward, skirts become more of an assumption--although even then women may make exceptions for exercise, etc.
Hair covering - Conservative women almost never cover their hair on a regular basis. If they do, it is often with a kippah. Within liberal Modern Orthodoxy, you will find a whole range, from women who wear a full covering such as a tichel on a daily basis, to those who wear a partial covering, to those who wear no covering, to a great many who may wear a covering only on Shabbos/to shul and for candle-lighting. Within mainstream and right-wing Modern Orthodoxy, it is more common to find women covering their hair full-time with either a full or a partial covering, although even here, there are significant numbers of women who cover only for Shabbos/shul and holidays.
In my experience, mixing with men at work is absolutely standard for all manner of Modern Orthodox women as well as all Conservatives.
Swimming pools are harder. In Modern Orthodoxy, women generally do swim in mixed settings. They will usually wear something a little more modest than a regular bathing suit. The closer you get to mainstream Orthodoxy, the more women will cover up for the pool, or else relegate themselves to single-sex sessions when possible.
One of the whole points of Conservative Judaism is full equality and integration of women — and all other sorts of people — in all aspects of religious and secular life. There are countless Conservative rabbis who are women. There are also Conservative rabbis who are gay or trans-sexual. People in our congregation wear whatever they want to wear but both men and women are encouraged to put on prayer shawls and kippot (yarmulkas) during services. Women who don't are encouraged to wear some sort of head covering but it doesn't have to cover the hair. A headband will do. So will a baseball cap. Modesty is typical for women during services but no one objects if someone comes to shul (synagogue) with a short skirt or plunging neckline. Outside of the synagogue, Conservative women are often at the forefront of social justice movements, particularly feminism. In my congregation, the only women who don't have jobs are either retired or disabled.