Covered vs. Non/Un-Covered Basis

[Dezrah]

Does covered versus non-covered basis have any real meaning if we're currently in the 15% marginal bracket?

Here's a quick summary of my situation:

I have an UGMA/UTMA account from back when my parents tried to save for my college.  They wound up cash flowing almost all of my costs so most of the principal has ridden the stock market roller coaster for 15 years.  For most of that time, our income was low enough that this account basically didn't affect our taxes at all.  Lately though, the fund managers seem to be churning a lot more and we earn enough that the stupid fund is requiring us to write a check at tax time.  When I couple this with the (relatively) high ER, I've decided it's time to get out and get it in a boring Vanguard account.

This tax year is kinda weird for us.  Our income is more than ever, but periods of unemployment still make this a low-income year, so we're still in the 15% bracket now but probably won't be for many years to come.  Is this the best time to move it or does it even matter?

[MDM]

Does covered versus non-covered basis have any real meaning if we're currently in the 15% marginal bracket?

No.  Covered vs. non-covered refers to whether your broker has to keep track of your basis or not.  E.g., see http://www.usfunds.com/investing-with-us/faqs/cost-basis-faqs/what-are-covered-and-non-covered-shares/

...we're still in the 15% bracket now but probably won't be for many years to come.  Is this the best time to move it or does it even matter?

Could be a good time if you would pay no tax on the capital gains.  It's called "Tax Gain Harvesting."  See https://www.bogleheads.org/wiki/Tax_gain_harvesting.