The future sixth-generation(6G) paradigm aims to seamlessly integrate communication and environmental sensing capabilities into a single radio signal, promising improved efficiency and cost-effectiveness through simultaneous data communications and environmental perception. At the core of this evolution, orthogonal frequency division multiplexing(OFDM) and its advanced waveforms emerge as pivotal for integrated sensing and communications(ISAC). This study introduces a concise and unified ISAC waveform design framework based on orthogonal multicarriers. This framework supports versatile applications of OFDM and its derivative waveforms within a generalized ISAC system, marking a significant leap in integrating communication and sensing capabilities. A distinguishing feature of this framework is its adaptability,allowing users to intelligently select modulation strategies based on their specific environmental needs. This adaptability optimizes performance across diverse scenarios. Central to our innovations is the proposal of discrete Fourier transformspread OFDM with index modulation(DFT-S-OFDM-IM). This framework is paired with newly proposed signal processing methods for single-input single-output and multiple-input multiple-output(MIMO) systems. Extensive evaluations highlight DFT-S-OFDM-IM's superiority, including dramatically reduced peak-to-average power ratios(PAPRs), competitive communication performance, and exceptional sensing capabilities, striking an elegant balance between communication capacity and environmental sensing precision.