The spelling without the apostrophe is correct. 
- "There were two other tributaries of the Rivanna below Milton in early times, though their names are never heard at present, Henderson's and Miller's Branches. Moore's Creek has been so called from the first. The same is true of Biscuit Run; but the names of its branches. Plum Orchard on the east, and Cow Branch on the west, have slipped from the memory of men. A small prong of Moore's above Biscuit Run once had the name of Edge's Creek; it is forgotten now." Edgar Woods, Albemarle County in Virginia: Giving Some Account of What It Was by Nature, of What It Was Made by Man, and of Some of the Men Who Made It (1901)
- The crossing of the Rivanna at the Free Bridge was known at the beginning of the last century as Moore's Ford.
For several years it has been known that Moores Creek in the urban portions of the community is impaired and is contaminated by fecal coliform. 
An eroding abandoned landfill on Avon Street Extended, formerly jointly operated by Charlottesville, Albemarle, and the University of Virginia, was a major contributor to contamination. In 2009 the developers of Fifth Street Station, Riverbend Development, partnered with Albemarle and the Department of Environmental Quality to begin remediation of the landfill site.
In 2012, the Thomas Jefferson Soil and Water Conservation District began offering cost-share assistance to help farmers and homeowners in the Moores Creek watershed pay for septic system repairs and conservation programs to help restore water quality.  Also in 2012, the Rivanna River Basin Commission received a grant of $267,000 from the state Department of Conservation and Recreation to remediate bacterial contamination over the following 2.5 years.
The Thomas Jefferson Planning District Commission is planning and constructing the 5th Street Trail Hub to serve as a trailhead for an enhanced trail heading west toward Azalea Park. This hub will be near the confluence of Moores Creek and Biscuit Run. 
- Web. Albemarle County In Virginia, Rev. Edgar Woods, The Michie Company, Printers, 1901, retrieved May 7, 2019.
- Web. Albemarle County In Virginia: Giving Some Account of What It Was by Nature, of What It Was Made by Man, and of Some of the Men Who Made It, Rev. Edgar Woods, The Michie Company, Printers, 1901, retrieved May 7, 2019.
- Web. [https://web.archive.org/web/20100709225018/http://www.tjpdc.org/pdf/rep_envi_mc_Prop.pdf The Fecal Coliform Problem in Moore’s Creek Watershed: Understanding the Sources, Finding the Solutions], Thomas Jefferson Planning District Commission, April 1999, retrieved February 23, 2012.
- Web. Hot dam: Should it stay or should it go?, The Hook, Dec 5, 2002, retrieved 2019-05-07.
- Web. Going down: River runs through old dam, The Hook, Aug 16, 2007, retrieved 2019-05-07.
- Web. Removing Barriers to Healthier Rivers, Appalachian Voice, April 1, 2005, retrieved 2019-05-07.
- Web. Woolen Mills Dam Breach, Rivanna Conservation Society, August 15, 2007, retrieved 2019-05-07.
- Web. Old Landfill May Get Clean Up, March 3, 2009, retrieved 2019-05-07.
- Web. Thomas Jefferson Soil and Water Conservation District Offers Cost-Share Assistance to Moore’s Creek Watershed Residents, County of Albemarle, February 14 2012, retrieved February 23, 2012.
- Web. Moores Creek Bacteria TMDL Implementation Project, Rivanna River Basin Commission, 2012, retrieved 2019-05-07.
- Web. Open house held for future trail hub near Wegmans, Sean Tubbs, News Article, Charlottesville Tomorrow, Charlottesville, Virginia, March 15, 2018, retrieved March 16, 2018.