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When you have had to use cash for all your purchases and have not really used a credit card until now it means you haven't really had a solid credit history. In order to get a credit card you need credit history and it is an important facet in many other financial situations. Having a loan or a mortgage is also a good way to build credit history, but the highest consideration is for revolving credit which comes from owning a credit card. The problem with having no credit score or history is that it is equated with having bad credit. Since you have been maintaining a good score by paying on time, you can consider little steps to building a credit history that banks will trust.
The first thing to do is to see what options are available to you. Since you have no definitive credit card in your name and seem to be running into dead ends trying to get one, you could start by looking to sign up for a store credit card. These will be for low credit limits and will not be difficult to obtain. It is a good way to start having some kind of records in place. Another option you can consider for getting a credit card is to consider opting for a secured card in the first place. With a secured card many people find it easier to build credit history and there are many secured cards which send in regular credit reports and help those with no and poor credit history become eligible for an unsecured card eventually.
The last option you can consider is to have a co-signor in getting your credit card. That way you can easily be eligible and obtain a credit card for your needs in no time. There are definite advantages and disadvantages to this. The advantage is that the other person's good score will give yours a boost, but if either of you default on payments, you will both suffer. So it's equal number of pros and cons. When you have eventually built up a good credit score, you can move onto considering a number of applications and mailers for unsecured credit cards in the long run. Things may seem difficult at first, but it can better, all you need to do is find out the right ways to approach the situation and tackle the issue.