[edit] Parks

[edit] Dublin Park

Located at 8324 Old Madison Pike, just east of Hughes Rd.
Includes an outdoor swimming facility with baby pool and diving well, a double court gymnasium equipped for basketball and/or volleyball with an upstairs walking track, a 25-yard heated indoor swimming pool, multi-purpose room, meeting rooms and administrative offices. A walking trail, five (5) soccer fields, a community built playground and seven (7) tennis courts are located within the 60 acre park.

There is actually a "dog park" in 5 pts under 565 by Pratt. Go by Olde Towne Coffee and people should be able to tell you how to get there. ~Reference Link
The state park is great! Absolutely bring your dog. :) I might also recommend the numerous trails of the Land Trust. Start with the back side of the mountain, the parking lot is on your right as you go up Pratt Ave/Toll Gate. And you can take your dog there too. :) ~Reference Link
I agree with the state park and Land Trust recommendation. There is a sidewalk up Bankhead Parkway that extends at least to across the street from the trailhead for Wagon and Tollgate trails, so you can walk your dog up there. The Land Trust and state park trails connect, so you call walk all the way to the bottom of the east side of the mountain. Does your dog have a particular critter/prey preference to see? Groundhogs are more common on Monte Sano. Bunnies are more common in Hays Preserve which is close to Hampton Cove. Deer are common in both. ~Reference Link

Located at 200 Cleveland Avenue, NW (behind the Historic Huntsville Depot, between Church and Meridian Streets near I-565) http://www.opencyclemap.org/?zoom=16&lat=34.73665&lon=-86.58844&layers=B000 CC BY-SA OpenCycleMap from OpenStreetMap data
a FREE venue open to the public from sunup until sundown; named for Lydia Gold, an area skateboarding activist in the 1980's.
http://worldconspiracy.org/sk8/index.php/content/view/56/70/