In this paper, a class of single-layered multifunctional leaky-wave antennas (LWAs) with flexibly engineered radiation and filtering characteristics are proposed and demonstrated for microwave and millimeter-wave applications. Radiating discontinuities (RDs) exhibiting multiple resonances while particularly possessing flexible model-control capability are exploited to accomplish such design freedoms and multifunctionalities of LWAs. By properly engineering the resonance characteristics of RDs under the mode-control principle, the attenuation constant of relevant LWAs can not only be freely tailored for diverse beamwidth/directivity requirements, but also simultaneously maintain a flat frequency response for radiation stability. Meanwhile, controllable filtering behaviors can be obtained as well by the LWAs thanks to the transmission zeros introduced by resonances. Consequently, both the radiation and filtering performances of LWAs can be adequately tailored by taking advantage of the mode-control capability of RDs. Under this design concept, two types of LWAs based on substrate-integrated waveguide and microstrip techniques are respectively developed for different system integration platforms. The substrate-integrated waveguide LWA whose unit cells consist of different longitudinal slots is firstly examined. Additionally, the microstrip LWA, which depends on stub-loaded resonators, is further investigated. The proposed two LWAs are all with flexible engineered electrical behaviors, single-layer, low-cost, and easy integration; they may be a potential candidate for various system applications such as 5G communication and Internet of Vehicles.