What are my options for renting an apartment / townhouse without a credit score?

Do you mean without a credit score? Because in the United States, very few landlords require a tenant to have a credit card. If you are suggesting you don't have a credit score/file and are thinking you need a credit card to get one, then you are on the right track. A revolving credit line, like a credit card, is a great way to build a positive credit history, but not the only way. You can apply for an installment loan, secured card, retail card, or even a P2P loan. Another alternative is to have a co-signer, someone with good credit, sign your lease. There are also many understanding landlords who, with proof of income, would be willing to rent to you.

Chances of renting an apartment without any established credit history are almost zero to none. If you're contacting an apartment complex, they always use a third-party software application that runs your credit and software app tells them if you're credit worthy.  If you don't pass the minimum requirements, they will reject you and ask if you have a cosigner with good acceptable credit. If the above option fails, you might want to find a private apartment owner who would be able to work with your somehow and get you in there.  However, the chance is also not that high. To establish legitimate credit, it's best to just open yourself a credit card, charge on it and pay off before it's even due.  Do that for 3-6 months and you'll see how your credit is going to get established and you score will surely be above 600.