I'll just give a quick rundown.
Monarchy in any kind of autocratic sense can be at best, a benevolent dictatorship and at worst, tyranny. No arguments in favour of tyranny here, for the most part it is something that people find intuitively wrong.
As for benevolent dictators, hereditary or otherwise, I've seen some people speak up in favour of these. I can really understand this viewpoint but for me, it isn't just enough for a country to prosper. It is a truly great thing when that country prospers as a result of the effort and direction of its citizens. Any dictator, however benevolent, if he is in control of the moral aims and values of any particular society; this amounts to paternalism. I personally don't want a country's goals, moral labour, legislation etc. directed by any particular individual, I want to direct these things myself along with my fellow citizens. Decisions are just inherently better when you make them yourself, human nature tells us this.
So that covers monarchs with any real power.
Then comes the old saying "Do you know how stupid the average person is? Well, half of them are even dumber than that!" As for those who use the counter-argument that the people are stupid and don't know what's good for them, I would respond that that assertion makes no difference. The decisions aren't good because they are the right thing (although I'm not sure how you could define 'the right thing' and which values you'd use); they are good because we, the citizens, made them.
Democracy isn't about making good decisions, democracy is about representation. Anybody who doens't realise that hasn't given it enough thought. What people in the western world value above all else isn't a better climate, or a national healthcare system (although a lot might, I'll come to this), its freedom, or liberty. Democracy is the best form of government because it maximises the freedom of the individual. Any other moral goals, such as a national health service or the war on terror, are pursued because they maximise freedom (e.g. a sick person possesses less positive freedom, as does a country plagued by terrorists).
As for people that think the system is broken somehow and came about by fluke or accident, I think they are failing to be impartial and approaching the debate with the wrong premises. To try to provide ridiculous legislation or events or monarchs as counter-examples is to go about the debate in the wrong way. You need to begin with the premise that the system works and then seek to explain how it works through things like moral values or pragmatism or intuition. In my opinion, and this is quite controversial, countries get the governments they deserve. America deserved George Bush, and what better way to deserve a government than by electing it?
Finally on to puppet monarchs such as the British Royal Family. I'll deal with these by way of example.
My personal opinion on this is that they have been reduced to a laughing stock, the fodder of the British tabloid press, with no legislative influence - indeed, very little influence whatsoever. Technically, they have serious power and authority in government, but if they ever tried to exercise it in any meaningful way they would be immediately deposed in short order, and I think they know this hence they haven't given it a shot for the last century or so.
However, even the diehard pragmatist would struggle to argue against their presence since their benefit to the British taxpayer far outweighs their cost. The cost very little in terms of national budget and they create far more value through public service and other things like tourism. I'm more than happy to let them do their thing, even if for the most part they embarass the UK (Prince Philip and the Chinese in Hong Kong, Prince Harry and the pot smoking etc.), in other ways they are extremely beneficial providing a focus for national heritage and identity and even in some cases, a few role models. Let's not be too tough on them, they're real people and I've done far worse without it being splashed all over the papers.
A lack of national identity or a sense of heritage is something that I believe America really suffers from. Indeed, it's not really something they can grasp and these ties between citizens, especially in a time when society is becoming more closed with cars, personal computers and the internet, I believe are worth hanging on to.
When I was at Oxford I overheard an American tourist in the Bodleian Library say to a librarian "The architecture is beautiful! Is this building pre-war?" and the library politely replied "Madam, this building is Pre-America". Things like the Bodleian Library, The Royal Family, the British sense of humour and chicken vindaloo all add up (although in an unquantifiable way) to formulate an answer to the question "What does it mean to be British?" And I think in removing our monarchy we'd be removing a part of ourselves. I'm sure its a similar case in countries like Denmark.
So the short version.
Autocracy is always bad.
Democracy will always beat autocracy hands down on the grounds of greater freedom.
Modern monarchs such the UK Royal Family are harmless, if not beneficial and should be tolerated.
Feel free to disagree.