So, this is not purely a question about political popularity in a vacuum thanks to the clarification you've added. Indeed, when you interrogate the specifics, it ends up kind of turning into two separate questions, one about the popularity of the Qing, and the other about empire and decolonisation.

With regards to the former, the answer is difficult and complex. There were no opinion polls assessing the popularity of the monarchy, nor any way of really definitively breaking down which regions were more or less pro-Qing than others. We can, quite fairly, say that at least among elite intellectuals – note, these were not a hereditary nobility as such – there was a broad opposition to the Qing monarchy as it had then existed, in the sense of an absolute monarchy unconstrained by any form of constitutional law, and operating with an implicit pro-Manchu ethnic bias. But this did not lead to a single uniform response.

Over the course of the 1890s and 1900s, in effect two political tendencies, not entirely in opposition nor entirely in concert, emerged: Republicanism and Constitutionalism. Constitutionalism does not necessarily mean constitutional monarchism; rather, Constitutionalists sought to establish the rule of law in China, with a government ultimately based on a binding constitutional document or documents that clearly delineated the powers and responsibilities of various branches of government, and which transcended any individual act of arbitrary power. This could be achieved under a monarchy, but also under a republic, and many Constitutionalist leaders of the later Qing carried over into the early republic with considerable influence. The most prominent of these figures was Liang Qichao, who had been one of the major leaders in the abortive Constitutionalist reform movement of 1898, and escaped to Japan after the crackdown against the reformers initiated by Dowager Empress Cixi. Republicans, on the other hand, were not necessarily committed to any sort of deeper philosophical basis of government and statehood; they mainly just wanted the Qing gone and a new, Han Chinese-dominated regime installed in its stead.

These two tendencies ended up appealing to slightly different groups. Both, of course, had considerable appeal among globally-connected intellectuals who wanted to see China 'modernise' along the lines of Japan, but when it came to penetrating further down the social scale, there were some differences. Republicanism became especially prominent in the Qing military, which was effectively the centrepiece of the empire's 'modernisation' efforts. But Constitutionalism, or dimensions thereof, managed to gain a certain broad-base appeal, most prominently in Sichuan, where 1910-11 saw considerable controversies and protests – eventually boiling over into the final set of crises precipitating the revolution – over the role of foreign loans in railway construction, amid a general spread of a sense of 'popular sovereignty' whereby those outside the traditional system of government demanded a greater say in its running. But that doesn't translate directly to agitating for the Qing to be overthrown.

The broad conclusion in recent historiography on 1911 – though with some versions of the argument going back to at least 1975, when Edward Rhoads' study of Guangdong in 1911 was published – is that the fall of the Qing was mostly to do with a series of disastrous, credibility-incinerating decisions made in 1911, rather than the unintended yet predictable effect of the New Policies instituted after the Boxer Uprising. As Rhoads and others have argued, the New Policies did successfully open up avenues for Han Chinese elites to participate more directly in government, and ultimately did succeed in creating elected provincial assemblies in 1909 and a national assembly in October 1910, although for the time being their role would be principally advisory. On the whole, by acceding to constitutionalist demands, the Qing effectively won over, or at least appeased, the elites agitating for political reform. When Sun Yat-Sen's republicans tried to launch a revolt in Guangdong in April 1911, it failed to win over much support at all and ultimately proved an utter disaster, and the Revolutionary Alliance was left in disarray for several months, undergoing a major schism.

Ironically, had the Republicans just waited a month, things might have gone differently. Zaifeng, the Prince Regent and father of the young Xuantong Emperor, had been courting controversy ever since November 1910, when he had declared that the Qing would not create a parliament with real political power until 1913, such that the National Assembly would sit for a full four-year term. Many elites, including a caucus within the National Assembly, called on the Qing to establish a parliament by 1911, which Zaifeng adamantly refused to commit to. Then, in December, Zaifeng refused to ratify a bill passed by the Assembly legalising queue-cutting and also making it mandatory for civil servants, police officers, and members of the military. In May 1911, he assembled a cabinet whose thirteen members included only four Han Chinese, compared to nine Manchus of whom seven came from the imperial Aisin Gioro clan, which proved to be one of the last straws on the camel's back when it came to the credibility of the Qing court's commitment to reform. While the immediate spark that lit the revolutionary fire would be a bungled bomb plot in Wuchang by a revolutionary cell in the army sympathetic to the Sichuan railway protestors, it was the failure of the Qing court firstly to firmly commit to political reform on a timetable desirous to the reformists, and then to apparently make a mockery of the whole process, which rapidly alienated not just reformist elites, but indeed to some extent a reform-minded public.

One thing I have generally only touched on in the above answer is anti-Manchuism, in part because it is somewhat hard to work into a relatively potted summary, and in part because it does deserve its own discussion. The simple fact is that the foreign origin of the Qing is an unavoidable aspect of late Qing political discourse. While many Constitutionalists believed that the Aisin Gioro clan could retain control of the throne under a constitutional monarchy, they nevertheless called for the dissolution of wider structures of Manchu-centric ethnic sovereignty. Some of an overtly Social Darwinist tendency, especially Liang Qichao, went a step further and advocated for the forcible miscegenation of Manchus with Han Chinese so as to subsume the former into the latter. More broadly, Han Chinese anti-Manchuism was rooted in the notion of Manchus as an 'inferior' people whose apparent position of dominance over the Han was aberrant not because any such ethnic hierarchy was unjust, but rather because the Han ought to be in charge over the Manchus and indeed all other minority peoples encompassed by the Qing Empire. The ethnic issue was thus one in which there was very little middle ground between Manchu privilege and Han supremacism.

So, were the Qing popular when they fell? The answer is, quite decidedly, no. The Qing had built up their goodwill among reformist elites in the 1901-1909 period by appearing to commit to a programme of political reform with the end goal of establishing a constitutional monarchy, but a combination of Zaifeng's own political intransigence and the unresolved – and potentially insoluble – question of Manchu status led to the Qing torpedoing that built-up credibility in 1910-11. The Aisin Gioro clan did not lose all of its supporters, even among elite reformists – for instance, Liang Qichao's mentor Kang Youwei publicly supported the short-lived 'Manchu Restoration' of 1917. But it was, at the end of its tenure, at a decided low ebb in popularity, even if not one that can be meaningfully quantified.