I've noticed in Bangladesh that one doesn't say Salaam between males & females - one to one that is. What exactly are the rules? Do they vary between Shia & Sunni? By social status?
Yes, men may give the salam to women, even if they are not their Maharim. The obligation, or the order to give the salam and return it is general for all Muslims. When salam is given to a women, or the other way round, it must be kept professional, there is no shaking of hands, and if the women is young and there is fear of Fitnah the rule changes. But in general, it is ok for men and women to give and return the salam if done professionally. May Allah forgive me of my mistakes.
For more information on this, see this (in Arabic).
In the name of Allah, the Entirely Merciful, the Especially Merciful
As a concise answer from the perspective of Shia Islam, in regards to saying hello (Salam) from a non-mahram man to a woman non-mahram and vice versa: All Maraja’ (Marja’al-Taqlids – religious authorities/scholars--): there is no problem (it’s permissible) if there is no intention of enjoyment (by that) and/or the fright (probability of…) falling in sin. Meanwhile, as a note: if the opposite gender is young girl, (then) it’s better not to say hello (Salam) to her in order to be safe from falling in sin.
Additionally, if I'm not mistaken, Sunni's view concerning that might be relatively/entirely so (similar to Shia view).
Can one Salaam between male & females?
Yes, if that male and female are mahram to each other.
No, if they are not mahram to each other. It will leads to fithnah as our Prophet (PBUH) said that
the adultery of the tongue is speech
-Sahih Muslim Book 033, Hadith Number 6421
What exactly are the rules?
Imam Nawawi (rahimahullah) states it straight clear and concise regarding this:
Al-Nawawi said in his book al-Adhkaar (p. 407): Our companions said: Women greeting women is like men greeting to men. But when it comes to women greeting men, if the woman is the man's wife, or his concubine, or one of his mahrams, then it is like him speaking to another man; it is mustahabb for either of them to initiate the greeting of salaam and the other is obliged to return the greeting. But if the woman is a stranger (non-mahram), if she is beautiful and there is the fear that he may be tempted by her, then the man should not greet her with salaam, and if he does then it is not permissible for her to reply; she should not initiate the greeting of salaam either, and if she does, she does not deserve a response. If he responds then this is makrooh (i.e. better not to do). If she is an old woman and he will not be tempted by her, then it is permissible for her to greet the man with salaam and for the man to return her salaams. If there is a group of women then a man may greet them with salaam, or if there is a group of men, they may greet a woman with salaam, so long as there there is no fear that any of the parties may be tempted.
Do they vary between Shia & Sunni?
No, it's common to all.