Shura (mutual consultation) is clearly prescribed in the Qur'an, as follows:
وَالَّذِينَ اسْتَجَابُوا لِرَبِّهِمْ وَأَقَامُوا الصَّلَاةَ وَأَمْرُهُمْ شُورَىٰ بَيْنَهُمْ وَمِمَّا رَزَقْنَاهُمْ يُنفِقُونَ
And those who have responded to their lord and established prayer and whose affair is [determined by] consultation among themselves, and from what We have provided them, they spend. [42:38]
In particular, are there any examples from the sunnah where the prophet used shura for appointing rulers?
As I have stated in my answer to this question, the Quran is largely silent on the details of how an Islamic government works. We do know (from the Shura ayah) that there should be some form of collaboration. We know that the government should be righteous. But other than that there is nothing.
How if we look at Hadith, we see the prophet consulting with his companions at times, and at times taking the decision alone. But, we know that the prophet was rightly guided by Allah. Are other people as rightly guided? That is a matter of debate and at the core of the schism between Sunnah and Shi'a, so I will not go into that.
Many scholars have interpreted this absence of clear direction to mean that humans have been given free reign to choose the form of government that they see fit.
