Due to its high lethality amongst the elderly, the safety of nursing homes has been of central importance during the COVID-19 pandemic. With test procedures becoming available at scale, such as antigen or RT-LAMP tests, and increasing availability of vaccinations, nursing homes might be able to safely relax prohibitory measures while controlling the spread of infections (meaning an average of one or less secondary infections per index case). Here, we develop a detailed agent-based epidemiological model for the spread of SARS-CoV-2 in nursing homes to identify optimal prevention strategies. The model is microscopically calibrated to high-resolution data from nursing homes in Austria, including detailed social contact networks and information on past outbreaks. We find that the effectiveness of mitigation testing depends critically on the timespan between test and test result, the detection threshold of the viral load for the test to give a positive result, and the screening frequencies of residents and employees. Under realistic conditions and in absence of an effective vaccine, we find that preventive screening of employees only might be sufficient to control outbreaks in nursing homes, provided that turnover times and detection thresholds of the tests are low enough. If vaccines that are moderately effective against infection and transmission are available, control is achieved if 80% or more of the inhabitants are vaccinated, even if no preventive testing is in place and residents are allowed to have visitors. Since these results strongly depend on vaccine efficacy against infection, retention of testing infrastructures, regular voluntary screening and sequencing of virus genomes is advised to enable early identification of new variants of concern.