I get cold easily, but don't like to wear long sleeves because they can fall or drag in poop. I also need shirts with modest necklines under my scrubs because of the V-necks of my scrubs show a bit of decolletage. These 3/4 sleeve boatneck shirts from Old Navy
are perfect for wearing under scrubs! They're lightweight, won't add bulk or make you too warm, my chest is completely covered, they're form fitting enough not to drag but still relaxed fit, the sleeves stay out of the way, and they come in tons of colors! I just ordered 6 of them in white, which means that I can wear them with any color scrubs! They're only $10 each, and Old Navy has frequent coupon codes for more discounts and free shipping!
I had some 3/4 length sleeve shirts from Land's End that I wore under scrubs, but they're starting to get holey after years of use. For now, my quest for perfect scrub undershirts is over! Perhaps I'll buy about 10 more to really make sure I'm covered.