It all depends upon the application. A DVM's perceived triple-digit accuracy can be misleading (unless you pay a lab to have the device calibrated annually) but few of us actually need that kind of accuracy anyway. And they do have some unique advantages, as noted by others.
I still prefer an analog mirrored scale Simpson 260, or a Triplett 630 for much troubleshooting work, although not inexpensive. An analog scale meter needle readily displays small variations that you'll not catch with a DVM; all you'll see there is a blurr of everchanging numbers. And ohm-ing semiconductor junctions, which is a breeze with an analog meter, becomes more confusing with a DVM.
For your tube biasing application I recommend getting an inexpensive analog scale meter; readily available from Radio Shack, etc. When you're tweaking those bias pots, the analog scale displays not only the absoulte value of your mA bias current, but you see how the tube is behaving with respect to stability. As tubes age & start becoming shakey, you can observe their instability much more readily with an analog scale vs. a the blurr of digital readout.
It's also handy having a budget model DVM on hand, & as they're now so inexpensive you might as well get yourself one of those as well while you're in the store.
And ensure your test leads have detachable alligator-clip ends, &/or use some separate clipleads, for those measurements when you can't or shouldn't be holding those probes (as in close proximity to those high plate voltages).