Are there any significant indicators that we believe empires or states to have exhibited prior to collapse? If so, is there an established methodology for understanding the decline of states, or do cultural differences between time and place render this too problematic?
I study modern Central and Eastern European history at the University of Kansas as a PhD candidate, and although I narrowly specialize in the history of technology in Czechoslovakia, I'll make an attempt to (tentatively) answer your question. The decline and collapse of empires is a big question in our field. The collapse of Austria-Hungary, Hohenzollern Germany, Tsarist Russia and the Ottoman Empire left discernible traces on the region in the late 19th and early 20th centuries, and the disintegration of the Soviet Union's power, and then its existence as a state, in the late 1980s and early 1990s had a similarly epochal effect. The consequences are still playing out.
Some thoughts on the characteristics of a declining empire (I hope others in this sub can augment or correct what I'll propose).
The empire should be attempting to reform itself; however, there are clear signs that reform has stalled in key areas, has not successfully addressed underlying problems, and that elites are deeply divided on the question with a strong reactionary cohort. The Ottoman Empire, Tsarist Russia and the Soviet Union are good examples of this (though the USSR's status as an empire is sometimes pejorative and usually debated, it should be noted).
The personality and characteristics of the monarch, emperor or empress can be key. The degree of personal power they've accumulated in their person means that weak, ineffectual monarchs (Tsar Nicholas II), monarchs obsessed with building a navy (Kaiser Wilhelm), older monarchs out-of-touch with present realities (Franz Josef) or well-meaning reformers unaware of the flood they're about to unleash (Mikhail Gorbachev), can have an unusually powerful effect in tipping an empire into decline in the first place or hastening the process along.
Tax revenues may decline or stagnate while costs (for modernizing a military, importing machine tools to industrialize the economy, paying off debt from past wars, etc.) should escalate, placing the state in a fiscal bind. This can stress the state's capacity to defend itself and limit expenditures on public works and improvements.
A notable short-term change in daily life takes place for ordinary people. Maybe the price of bread suddenly escalates. Maybe the army cracks down on protesters in a way they hadn't before, shooting and killing some this time. There are shortages of staple goods; people had gotten used to long lines and rationing for other things before, but there had always been potatoes and beer. But suddenly bare shelves.
Scandals at 'the top' the people gossip and joke about, revealing corruption or moral foibles at the top. Nothing that causes revolutionary outrage, but which breaks the myth and magic of monarchy. I'm thinking particularly of the allegedly perverse sexual access Grigoriy Rasputin was said to have enjoyed to the Tsarina and her lady-in-waiting during World War I while Nicholas II was away at the front, but other examples from the other empires exist.
Military defeats. The fundamental responsibility of an empire, like a state, is to protect its subjects/citizens and, often, to project power in a way that burnishes the reputation of the empire and subjects' 'buy-in' and pride in belonging to the empire. The USSR in Afghanistan, the defeats in WWI that toppled Germany, Austria-Hungary and the Ottomans, the poor performance and humiliating defeats the Russian Army suffered in Galicia and Poland - they all strike a blow at the prestige and competency of the empire.
Dissatisfied fringe elites. I'm making up that term for now. Essentially, empires are by nature multi-national, multi-ethnic, multi-religious, etc. They're multi-. And therefore, while they might have a 'core' group (or even two - see Austria-Hungary!) they also rely on fringe elites in the Baltic republics, or in Bohemia, or in Palestine, to help keep the peace and in-flow of tax revenue at a reasonably low cost of administration. Given that another characteristic of empires in decline might be fiscal problems, empires in decline are uniquely vulnerable to fringe elites' desire for increased autonomy and control of tax revenue/political independence for themselves. Hence Tsarist Russia and the Soviet Union saw a somewhat similar pattern unfold in 1918/1990-91 as outlying regions, responding to crisis in the imperial center, began to withhold tax revenues and assert independence, which only further weakened the empire and deepened the crisis at the center, which further stimulated actual autonomy in the periphery in a downward spiral of rapid decline.
Anyhow, that's all I can think of at the moment. It's fairly late and I have to teach sections tomorrow (this!) morning. I hope this addresses your question substantially at least in part. Other, smarter folks with more time on their hands will ideally jump in as well. Below is a reading list of some of the sources that inform my thinking in this post. Theories on revolutions themselves are also complex and interesting - I've recently been reading Padraic Kenney's "Carnival of Revolution" and trying to incorporate it in my own research, for example.