What is the ruling on demonstrations? What are the evidences for its permissibility? And what are the evidences for its prohibition?
The issue of demonstration is a long one, and I will summarize my words regarding it by saying:
Some of the brothers that said that demonstrations are permissible researched in the Sunnah after evidences for this, so they mentioned for example the famous hadeeth of Abu Hurayrah regarding the neighbor, something which is these days referred to as "rousing the public opinion", the source for this hadeeth is in Sunan Abi Dawud.
And also they use the hadeeth of Abdullah bin Iyas bin Abi Dhubaba where women complained (in front of the Prophet (ﷺ)) regarding their husbands (i.e. a female demonstration), and this is also narrated in Sunan Abi Dawud.
But all of this isn't necessary (to prove it's permissibility), because the rule is: "Nothing is legislated in this Religion except what Allah has legislated, and nothing becomes forbidden except what Allah has forbidden." So the rule is regarding worship ('ibadah) is that of tawqif (i.e. not permissible to exceed beyond), so there's a must to provide an evidence for it.
And the rule regarding customary things ('adaat) is that they are in general permissible. So whoever forbids something is asked to provide an evidence (daleel) for it, this is the fundamental rule (asl) regarding this.
So whoever permitted demonstrations and holds to this fundamental rule, then he is of no need to bring forth any evidences, because nothing is haram except what Allah has made haram. And even if one brought forth evidences to prove this, then it's being generous. So what's left for us is to see what those that forbid demonstrations have for evidence, then afterwards we can answer them and let the issue be finished (over with).
The most famous evidences that they have are three: That it is an innovation (bid'ah), that it is imitating the disbelievers, that it is forbidden as a means to block an evil (sad al-dhara'i) which leads corruptions (fasad).
1) With regards to their argument that it's a bid'ah, then this is false (batil). Because bid'ah is in matters pertaining to worship ('ibadah), not customary things ('adaat). And thus there isn't any need for redundancy of words to show that this argument is void.
2) With regards to their argument that is imitating the disbelievers, then this is also false. For history is filled with people gathering and marching and demanding something. And go back to the history of Uthman (رضي الله عنه) and go back to the history of al-Basrah and al-Kufa in the latter half of the first century (hijri), and see how plentiful these sort of demonstrations were. This isn't to prove that its something legislated or permissible, rather it is evidence that it isn't from the issues of imitating the disbelievers.
3) And regarding their argument that its something that leads to corruption (fasad), then this is not correct. Because we see it taking place everywhere and there is no corruption that takes place like which they claim will come about.
Rather great benefits (masalih 'adhimah) have come through it, like the overthrowing of the Egyptian taghut (i.e. Mubarak). And his overthrowing - even though it didn't bring forth a Islamic governance - eased the injustice and tyranny, and it brought forth some justice that weren't there during his (i.e. Mubarak) time.
This is a quick summary regarding this issue, and Allah knows best.
The source for the Fatwa is Fatawa al-Ha'iriyyah, a compilation of fatawa by the Shaykh from inside the prison of al-Ha'ir, KSA.
 Full hadeeth:
Abu Hurayra said, "A man said, 'Messenger of Allah, I have a neighbour who does me harm.' He said, 'Go and take your things out into the road.' He took his things out into the road. People gathered around him and asked, 'What's the matter?' He replied, 'A neighbour of mine injures me and I mentioned it to the Messenger of Allah, may Allah bless him and grant him peace. He told me, "Take your things out into the road."' They began to say, 'O Allah, curse him! O Allah, disgrace him!' When the man heard that, he came out to him and said, 'Go back to your home. By Allah, I will not harm you.'"
 Full hadeeth:
Iyas bin 'Abdullah (May Allah be pleased with him) reported:
Messenger of Allah said, "Do not beat Allah's bondwomen." When 'Umar (May Allah be pleased with him) came to Messenger of Allah and complained saying: "The women have become very daring towards their husbands," He gave permission to beat them. Then many women went to the family of the Messenger of Allah (wives) complaining of their husbands, and he (the Prophet) said, "Many women have gone round Muhammad's family complaining of their husbands. Those who do so, that is, those who take to beating their wives, are not the best among you".