yurta wrote:But how can I find the one that's likely to give the highest limit?
As indicated above, it's not just the card it's your credit and income as well.
yurta wrote:I've done a little research on the credit karma website, and it seems that average limits for people with my score are about $7K. Does that sound right?
No idea as I haven't done a survey of a statistically meaningful sample myself but take anything from CK with a grain of salt. Also, it's not just your score that determines your limits. With high utilization I was only able to get $2K on a Slate. Dropped it down to under 10% and picked up a couple of $25K cards. I didn't pull my FICO's at the time but utilization and 5 months time were the only differences between the applications. Not saying your credit supports this -- just pointing out that factors such as utilization can really hold you back.
yurta wrote:Can I then apply for two cards at the same time, hope I get accepted, and that my credit score doesn't get bashed too much?
Could go either way. It's impossible for us to say especially without knowing the details of your credit. With a 780 odds are probably pretty good that a couple of apps won't hurt much but, again, it's not just about the score. Consider your existing inquiries, their age, AAoA, utilization -- in other words, all the usual suspects that factor into credit.
yurta wrote:Maybe I could ask if they would increase it, although I doubt they'd agree to a factor of 10 increase, and I'd prefer a 0% card anyway.
Any CL will probably help. Remember that utilization = balances / available credit. More available credit will reduce your utilization.
Having only one card doesn't help, as stated above. Not using your only card also doesn't help. You want more than 0 utilization reporting. It will probably be difficult to jump from $1K in available revolving credit to $15K in less than a year. I'd suggest looking into AmEx and the 3x CLI as it may help you to build larger credit limits. This is a process that just takes time.