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It is an urban legend that there is overwhelming evidence that odd perfect numbers do not exist. At least I think it's an urban legend because I haven't been able to find the overwhelming evidence. If you know of overwhelming evidence not discussed here, please email William at Odd Perfect dot Org.
I'm aware of two arguments that odd perfect do not exist: Sylvester's "web of conditions" and Pomerance's Heuristic. Sylvester's web is discussed on this page, while Pomerance's Heuristic is discussed on the Pomerance page. Sylvester's web, at least as usually described, implies a constant difficulty that will eventually fall to higher number searches. Pomerance's heuristic indicates there should be no large perfect numbers — neither even nor odd. Pomerance makes arguments for why the heuristic could be wrong for even perfect numbers but right for odd perfect numbers. Although I can understand some people finding the Pomerance Heuristic a "strong argument," I contend that such a conveniently nuanced belief cannot be judged "overwhelming evidence."
According to Gimbel and Jaroma, "much of the modern work done on the subject of odd perfect numbers has as it roots, the series of papers produced by Sylvester in 1888." James Joseph Sylvester is famous for his eloquence, and his 1888 statement on the unlikeliness of odd perfect numbers is often quoted:
This quote is usually followed by a modern list of the conditions which hem it in on all sides. For example:
There is great hope because nothing on the list that scales badly with the size of N. Given only these constraints, if the probability a random 500 digit number is an odd perfect number is "p", then the same probability applies to random 1000 or 1,000,000 digit numbers. It doesn't matter how unlikely the list is because more candidates will always overwhelm that unlikelyhood. If the probability is one in 10^{1000}, then there is probably an odd perfect number with 1000 digits. If the list is so hard that it's one in 10^{1,000,000}, then there is probably a million digit odd perfect number.
The traditional roadblock to this difficulty is the need to factor thousand digit or million digit numbers. Our search innovation is to search the subset that has every component small enough to factor. This process can continue until the sheer number of known factors becomes too large to manage. It seems reasonable to hope that our search will overwhelm the list before the data management threshold is reached.
Although an eloquent testimonial to the difficulties of hand computing an odd perfect number, from the modern perspective eloquence is the only thing Sylvester's web has going for it. Personally, I prefer this quote (in translation) from Descartes: