The formula for taxable social security, if you are married filing jointly, is to take half your SS benefit and add your other taxable income. If that totals more than $32,000, then some part of your benefit is taxed. At the bottom, it's 50% of your SS benefit, but it can go up to 85% of your benefit.
This is how SS taxation interacts with your standard deduction:
Suppose your SS benefit is $2000 per month or $24,000 per year, and you withdraw $50,000 from an IRA. Since half your SS plus your IRA income is more than $32,000, then your SS benefit is taxable. So then we take 85% of the SS benefit and add it to the IRA, so your adjusted gross taxable income is $70,400. Then you subtract your standard deduction which makes your taxable income $46,000. That's in the 12% bracket for married filing jointly (and the first $9000 is taxed at 10%) so your tax is about $5340.
California does not tax social security but it will tax your IRA.
If this was a Roth IRA, it would not be taxable income, so it would not make your social security taxable either. Or to put it another way going back to the formula, if your benefit was $2000 per month ($24,000 per year), then you can withdraw up to $20,000 from a traditional IRA without triggering SS tax. And the IRA money would be tax-free as well since it would be under the standard deduction. If you could find a tax-free source of income to supplement your IRA (like a Roth IRA or tax free muni bonds) then you might never have to pay income tax again.
