Learned joint source-channel coding (JSCC) enables robust wireless image transmission by jointly optimizing the transmitter and receiver over differentiable channel models. For bandwidth-limited and time-varying visual links, a single model should support user-adjustable transmission rate and adapt to changing wireless channel conditions, while also dynamically allocating resources according to spatial content. Existing content-adaptive or dynamic allocation schemes often rely on entropy coding, context/probability prediction, explicit rate maps or masks, or auxiliary allocation networks, complicating the encoder-decoder pipeline and increasing side-information overhead. We propose TS-JSCC, a single-model adaptive JSCC framework with tail-structured sparsification. First, an L1-based tail-structured sparsification objective encourages each token to retain an active feature-channel prefix while suppressing trailing ones. This enables content-adaptive feature-channel allocation with compact side information through active-prefix transmission. Second, lightweight stage-wise neural regulating modules use a normalized sparsity-control coefficient and the channel signal-to-noise ratio (SNR) to rescale intermediate features for single-model transmission rate and SNR adaptation. Experiments on CIFAR-10, Kodak, and CLIC2021 under additive white Gaussian noise (AWGN) and Rayleigh fading show that TS-JSCC achieves strong rate-distortion performance against the latest learned-JSCC baselines and remains competitive with the considered idealized separation baselines, while retaining a simple one-shot encoder-decoder without extra structures or computations.

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