I'm 39 weeks pregnant, and considering an epidural. However, I'm concerned about spinal headaches. Should I be?

No. A spinal h/a usually is from leaking of spinal fluid. It can happen but is treatable. Remember that this is a great time in your life and when the time comes you will make the right decision about the spinal or not. Ask you OB about the options and what they will do in the event something happens. You will be fine. Congrats.
Epidural. Spinal headaches are unfortunately a complication of epidural anesthesia. While not an anesthesiologist, rest assured every attempt is made not to cause a spinal headache but even in the best hands they occur. I would discuss this with your doctors - and plan accordingly. A planned procedure is certainly less risky than a "rush job".
Yes, but. statistically chances are in your favor: the risk of getting postdural puncture headache usually less than 5%, close to 1-2%. The actual incidence is determined by your anatomy, skills of the person performing an epidural and quality of the instruments. Although it can be very unpleasant, it is practically always self-limiting, i.e. goes away by itself in 7-10 days and can be treated if severe.
Yes and no. There are inherent risks with epidural catheter placement during labor and one of those risks is a spinal headache. This is an uncommon complication and it is typically a self-limiting complication and it is treatable if necessary.