There are plenty of short term apartment rentals clustered around Barrio Santa Cruz and Arenal quarter. Sevilla5 and veoapartments are local specialists in vacation rentals.
The area of Las 3.000 viviendas, a gypsy neighborhood, is a bit problematic.
The rest of Seville is very safe, including the center of Seville and also Triana. Just be aware that there are a lot of young people drinking in the Alameda nightlife area especially on weekends; so expect some disorderly behaviour. Other than that, nothing more.
Luxury apartments and villas rental in Seville
Genteel homes apartment agency probably has the best listings for upscale apartments in Seville. All their apartments are spacious, well equipped, and are centrally located. Many apartments listed are penthouses or have terraces.