With the exception of some very serious offences (which automatically bar a person from entering the USA) the process is discretionary, so nobody can say for certain which offences will prevent a person from obtaining a visa or whether the chances of getting a visa are improved as time goes by. However, somebody recently mentioned (here on AB) that their partner had been refused a visa for 2 offences of driving without insurance, so it's obviously taken seriously by US officialdom.
As Norman is usually the first to point out though, the US authorities have no direct access to UK criminal records, so many people simply 'forget' about their convictions and enter the USA (illegally) under the Visa Waiver Program.
Chris
PS: Although your 3 points for speeding have to remain on your licence for 4 years, they cease to count for 'totting up' purposes after 3 years, so they might already be 'dead':
http://www.direct.gov.uk/en/Motoring/DriverLic ensing/EndorsementsAndDisqualifications/DG_402 2550
PPS: I've just re-read your post. Although you might be able to enter the USA as a tourist, I doubt that you'd be able to acquire a residence visa for the USA.