Within the city of Dallas problem areas include Pleasant Grove, South Oak Cliff, South Dallas (Fair Park), anything near Forest and Audelia in Northeast Dallas.
You would be fine in most of the suburbs, but remember that the posts on here favor the northern suburbs and there is a perpetual myth that the only nice communities are located in that area. They are nice, but ARE NOT the only nice, safe areas in the Dallas-Fort Worth area. There are decent places to live all over. The suburbs south of Dallas like Cedar Hill, DeSoto, Lancaster, etc and cities like Grand Prairie are looked down on by a few people posting here, but none of them by any means are places to completely avoid. If they were truly that dangerous, I doubt that they would have grown by 35 to 45 percent over the last decade and continued to attract middle-class families of diverse backgrounds. This is coming from someone who has lived here for years without any issues.
To back up my claims, I consulted the FBI's latest Uniform Crime Report for statistics. The overall violent crime rate for the DFW area was 358.4 crimes per 100,000 population in 2011. Murders, rapes, robberies, and aggravated assaults are considered violent crimes. Big cities like Dallas (681.1) and Fort Worth (603.7) exceed those figures, but have a variety of neighborhoods ranging from bad to very safe. Other suburbs with high rates of violent crime include Balch Springs (788.4), Forest Hill (a Fort Worth suburb; 618.3), Everman (a Fort Worth suburb; 593.1), Addison (585.1), Arlington (496.9), Hurst (a Fort Worth suburb; 464.3), and Terrell (427.3). The mid-sized and large suburbs to the north, south, and east of Dallas had rates that were lower than the metro average.
The overall property crime rate for the DFW area was 3498.5 crimes per 100,000 population in 2011. Burglaries, larceny-theft, and auto theft are considered property crimes. The property crime rate for Dallas (5057.9) and Fort Worth (4640.2) again exceed those figures, but it can vary significantly from neighborhood to neighborhood. Other suburbs with an elevated property crime rate include Addison (6736.2), Balch Springs (6290.5), Terrell (5046.8), Hurst (4997.0), Arlington (4611.8), Mesquite (4582.3), and Duncanville (4047.3).
Depending on your personal level of comfort, you may or may not want to avoid any of the places listed above. However, I would recommend that you visit any area you are interested in before deciding to move there. If you plan on commuting to work, I would suggest living as close to your job as possible to avoid traffic congestion.
Crime in the United States by Metropolitan Statistical Area, 2011
Crime in Texas by City, 2011