From your photos it looks like it may be japanese stiltgrass, an invasive weed. See the attached link from Weeds Gone Wild for more information and photos
 http://www.nps.gov/plants/alien/fact/mivi1.htm
mh

It seems that a few characteristics of this growth may not be adding up to Japanese stiltgrass. For one, the roots are anything but shallow. The listed reference says that they should be easy to pull; the plant  in question is impossible to pull by hand. Second, the foliage, to date, does not seem to be exhibiting any winterkill. We've had light frost a few days ago, but even today's hard frost in Central Montgomery County - with lows of 27 in our garden - has left it looking completely unfazed.

I'm still watching and waiting on the assumption that it may be Japanese stiltgrass, but if it is not, what other possibilities are there? If it is not terribly invasive, it is actually looking good in this spot in the garden, and I'd leave it alone.

Did you plant this plant? Is this a new house and the plant has been established? Did it just pop up?
From your additional information the plant does not sound like japanese stiltgrass.The underground runners sounds like it may be a type of bamboo. There are clumping and running types.   We do not recommend planting running bamboo as the root system is invasive and very competitive with surrounding plants. Some municipalities have restrictions on planting and maintaining it. Bamboo is best confined to a container where the root system can be contained. See our publication on bamboo http://extension.umd.edu/node/6211
mh

I inherited it from the previous owner when I purchased the home in July 2014. This is a 1966 home. Since then I've had to mow back a few shoots popping up on the nearby lawn,  but they didn't seem to be growing or spreading very quickly. I didn't check for root barriers, though. 

We cannot say what type of bamboo you have.  You can monitor the growth and see if you want to maintain or remove it. If you want to keep the bamboo, consider installing a barrier to keep it in bounds. Google 'bamboo barrier'. Dig a trench at least three feet deep and place an impervious barrier like heavy butyl rubber. See our video on bamboo barriers http://www.youtube.com/watch?v=lx6WXjnO24Y&feature=c4-overview&list=UUwqjsyyQpwZ-HhcFE9L6ung

For information on chemical control see our bamboo publication http://extension.umd.edu/node/6211
mh