Are there any specific teachings related to how Hindus should dress in the public? In Islam, for instance, it has been outlined that women must wear a hijab or nikab. Islam has very clear instructions when it comes to clothing. Do Hindus have clear instructions on how, what, when or what not to wear based on special or non-special events?
I will try to answer your question. It is a valid but difficult one and I do not see any valid research that I can cite.
Due to the fact that Hinduism is probably the oldest "Religion", it was not developed by a single teacher or preacher. It originated from ancient India's way of living. Various unknown "Teachers" have contributed to this.
Do's and Don'ts never existed before its development and it was a free run in the social setup. The development of Do's and Don'ts started posing restrictions in the life of men & women mainly to avoid any intentional or unintentional 'hurt' to others. These rules and regulations categorized people through development of caste system (largely based on their profession) and hence form a society. This was a part of social fabric development that was later named as Hinduism. This process has been well described in the book the 'Discovery of India' by Jawaharlal Nehru.
The caste system imposed do's and don'ts according to the profession taken up by people. The attire they wore was based on the requirements of living to cover up themselves and their profession. If you see in ancient depictions and texts a lot of importance was attached to jewellery along with the clothes which was an integral part of clothing. In most cases the clothes remained as single woven piece of cloth (Dhoti or Sari) to cover the bottom half. An additional piece for women to cover the top half or wrap the same piece to cover the top too. The kings and the warriors (Rajputs) wore jewellery, weapons and other protection equipment as part of their clothing attire.
As far I know there is no clear definition or restriction on clothing in Christianity too. I am not sure how much my reply answers your question but I would strongly recommend you to read the book mentioned above for a deeper insight or watch an old series 'Bharat Ek Khoj' telecast on doordarshan earlier and now available on the youtube for a good understanding.