The most common visas are (not being technical as there are multiple types of visa under each category):
E visa---investor or essential employee of the company whose business with USA is a significant portion of its sales. For E, you have to be a citizen of the country in which your employer's parent or mother company is located. Example: if you are going to work for an all American company and your citizenship is in UK or Poland, this E visa is not for you.
L visa---transfer employee from the foreign company to its USA branch
H1B visa---college graduate with technical skills (Maximum term: 3 year + 3 year; you need to go home for 1 year and repeat 3 + 3 again; also, there is a annual cap; new year starts in Oct)
There are other visas for professional athletes (extraordinary skills), spouses, immediate families, and etc.
Once you get the work permit for the permanent residency status, you have to wait for the long waiting line. The recent amnesty US govt gave to illegal Mexicans have inundated the immigration system to the max and all the legal immigrants are pushed out because of that amnesty. The current Mexican back log will clog up the system for another 3 to 4 years at least, as those cases associated with that amnesty will take up the permanent residency (aka, green card) visa allocations.
In a nutshell, sponsorship means whoever sponsors you have the financial resources to do so. If a company sponsors you for H1B, E, or L, it has the financial resources to hire you. If you work for a small company, that company may have a difficult to show a satisfactory financial resources to get you a visa you need to work for that company. The bigger companies have no problem with L or E visa mostly.
Even if you cannot get the visa for USA, it won't be the end of the life. There are many other countries where you can bring out your skills to full use. There are too much of unsubstantiated hype about living in USA. I came here under the influence of that hype and I am now looking for some other country for my next career now. I wish I can trade mine (USA) with your position (UK).