Got my retirement visa last week. Was not asked for a criminal record report. Had to produce copies of passport (name/ photo page, visa page, page with latest entry stamp, departure card), letter from UK Embassy confirming address and pension earnings > 65000 Baht/ month (= alternative to 800,000 Baht in Thai bank account), copies of house book. You first have to register with the Embassy. The UK Embassy letter did quote that they had "no information to the detriment of the applicant" - maybe that is the polite terminology for not having a criminal record. As pointed out above - if you're from the UK then most crimes are 'spent' after a certain period and need not be declared and are not disclosed - in the UK! However this is not UK - if you do not disclose on the application form then you have (technically) given false info and they can use that if they need an excuse to cancel your visa. (Bit like asking on the visa application form for the UK - are you/ have you been involved in terrorist activities? Noone's going to answer Yes so if you're caught out all they need do is throw you out of the country for making false application/ declaration. No need for messy stuff like deportation orders etc!!)
I had a 12 month Non Immigrant O visa from UK and the retirement visa is an application for extension to stay for a further 365 days. If I remember correctly you can only get a retirement visa here in Thailand. Visas other than Non Imm O are ok but application cannot be made with the 30 days on arrival visa.
Good luck! Oh - and check first with the office where you make application as to copy docs required. Some have their own ideas. Mine was obtained in Hua Hin (& very helpful they were too)