Serious historians don't give it much credence in and of itself, and as a rule of thumb it tends to be invoked as a position to set oneself in opposition to. See, for example, Julia Lovell's The Opium War for a particularly overt example of this trend against the invocation of the concept.
The development of opinions on this matter has been a gradual one, and the popular understanding still lags behind the academic sphere to a considerable extent, as evidenced by a lot of past questions on this sub (not, of course, that it's anyone's fault, and it's encouraging to see that people are eager to learn more). The precursor to the Century of Humiliation was simple 'National Humiliation', a concept that emerged in 1895 with the defeat to Japan and which continued to play a major part in the decades to come, such as in the Railway Rights Recovery Movement that helped spark the 1911 Revolution and the May Fourth Movement in response to the Treaty of Versailles in 1919. The 'Century of Humiliation' as a discrete period of time, however, was codified by Mao, who portrayed the period of 1839 to 1949 as a period in which the Chinese nation was continually humiliated by foreign powers and its own 'feudal' landlords, and repeatedly foiled in its many attempts at resistance, until the Communists liberated the country from both. Mao's 1938 piece on military theory On Protracted War, alludes heavily to this, and the key acts of resistance remain memorialised on the Monument to the People's Heroes in Tiananmen Square, these being:
Before I address the fundamental problems of this narrative, however, I will conclude my lay about the history of the concept in Western historiography. Concurrent with Mao emerged the 'Harvard School' of Chinese history in the US led by John King Fairbank, which argued that the conclusion of the Opium War in 1842 represented a fundamental break in the history of China, and focussed its studies on the post-1842 period. As Paul A. Cohen later described it, from the 'Harvard School' came two main tendencies: the 'impact-response' approach championed by Fairbank saw modern (that is, post-1842) Chinese history as a succession of European actions and Chinese responses; while the 'modernisation' approach exemplified by the work of Mary C. Wright and Albert Feuerwerker saw it as a deeper cultural struggle between somewhat nebulous ideas of 'tradition' and 'modernity'. While not exactly parroting the 'Century of Humiliation' narrative, not least in that the 'impact-response' framework was a largely (if artifically) neutral one and the 'modernisation' narrative was one which held an underlying assumption about the superiority of European 'modernity', all three narratives shared a fundamental Eurocentrist bias about the trajectories of societal development and about China's place within it, and all saw violent contact with Europe as a hard turning point.
Two challenges faced the 'Harvard School', from opposite directions. Influenced by Maoist rhetoric and provoked by the imperialistic excess of the Vietnam War, in the 1970s what Cohen calls the 'imperialist critique' attacked the 'Harvard School' for sacrificing objectivity for neutrality, and downplaying the deleterious effects of imperialism upon the Chinese people. The 'imperialist critique' was, however, no less Eurocentric than its forebears in expondiing the 'Century of Humiliation' narrative as envisioned by Mao. What really overturned the 'Harvard School' and the 'Century of Humiliation' was a move towards a more Sinocentric viewpoint, begun on a small scale by Jonathan Spence, Joseph Esherick and Philip A. Kuhn, solidifed by Paul A. Cohen and accelerated by the 'social history turn', as history moved from being concerned primarily with the high politics of court, bureacuracy and military and began looking at ordinary life at the time. Increasingly, it was becoming clear that for most ordinary Chinese people, there was little to distinguish 1843 from 1838, and even Fairbank would later begin his surveys of Chinese history in 1800 instead.
Developments since then have led to further elaborations which have further nailed the coffin of the 'Century of Humiliation' narrative: the 'China-centred' generation of the 1980s generally disputed the idea of the Han-Manchu idchotomy as a fringe position that mostly died out after the first couple of decades of Qing rule, and advocated for seeing the Qing as part of a broader 'Late Imperial' phase of Chinese history, beginning no later than the last century of Ming rule in the 1500s (the arch-example being Jonathan Spence's The Search for Modern China). However, the 'Inner Asian turn' of scholars like James Millward in the later 1980s led to a major re-evaluation of Manchu identity at the turn of the millennium, culminating in works by Pamela Kyle Crossley, Mark C. Elliot and Edward J. M. Rhoads which pretty conclusively proved that Manchu identity was integral to Qing imperial ideology. 'Century of Humiliation' rhetoric had traditionally asserted the Sinicisation of the Manchus as part of its narrative, and so the academic consensus has since become even more alienated from the Communist Party line. Occasional challenges have cropped up within China, most notably Mao Haijian's groundbreaking revisionist history of the Opium War in 1995, but the post-Deng Xiaoping state has increasingly clamped down on criticism and revisionism, so most of the developments in the historiography of modern China have had to come out of Western institutions.
As regards the problems of the 'Century of Humiliation' narrative itself, the main criticisms boil down to four key points:
1: Whose humiliation? Conflations and constructed homogeneity
The 'Century of Humiliation' narrative rests on conflating failures of the Qing government in its international affairs with the humiliation of the people of China writ large. However, the microhistorical approach of Mao Haijian has shown that ordinary Chinese citizens and even much of officialdom were largely apathetic to the First Opium War when it was ongoing, even those in the warzones themselves, and were more concerned with their local interests than any broad national unity. It fails to see internal cultural divisions in China, particularly between Han and Manchu but also between Han subgroups, although in its original Maoist form it did acknowledge class divides (much diminished in modern rhetoric), and thus further projects a false sense of national homogeneity.
2: Projection
The narrative also rests on projecting attitudes, motives and interpretations of later periods back onto the Late Qing. The two main sub-points are:
2a: Retrojections of national identity
The 'Century of Humiliation' presumes from the get-go the existence of a unified Chinese national identity that had a strong reaction to foreign imperialism, and which eventually established a Han Chinese nation-state. However, such an identity did not begin to coalesce until the 1880s, in contact with Western notions of ethnic identity and in response to the increasing failures of the Manchu state. This is how the narrative gets round the state-people divide for the period of the Opium Wars – by denying its existence.
2b: Projection of anachronistic motives
If you remember the list above, many events seen by Mao as acts of the 'People's Heroes' were clearly directed aganst the Chinese government and mainly over domestic affairs, rather than against Western imperialists: the Taiping uprising, the 1911 revolution, even the Nanchang Uprising and Chinese Civil War. Movements with domestic motives are inaccurately made out to have anti-imperialist ones.
3: Problematic periodisation
If you look at the list of events on the Monument to the People's Heroes, the period 1851-1911 is notably bare. As I have often emphasised, the narrative breaks down considerably if you consider the period between 1862 and 1884, in which Western governments were actively bolstering the Qing state against its internal enemies and sponsoring its military and economic modernisation, and even then only France stopped doing so after 1884, and it was not until 1895 that China once again faced a serious outside threat.
4: Excessive focus on military disaster
The narrative rests mainly on looking at military defeats, but for much of the period, China's economy was mostly safe from imperial incursion until the railway loans and the Scramble for Concessions. As noted, until 1895 the prevailing mood was relatively ambivalent about imperialism, and whether there was much anti-imperialist sentiment in the early 20th century outside the educated classes is still quite questionable. By focussing on military affairs, the narrative unduly emphasises high politics and overlooks actual Chinese attitudes. Even then, certain high political affairs are ignored, such as the the 1898 constitutionalist reform movement and the Self-Strengthening Movement.
In short, the 'Century of Humiliation' narrative only works by ignoring inconvenient information and generally projecting anachronistic attitudes on what it doesn't. Oh and also,
5: It's Communist Party propaganda that is frequently revised to suit new political contexts.