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An $L^1_{loc}$ function on $\mathbb{R}^n$ is in $BV_{loc}$ iff its distributional derivatives $\partial_i f\in\mathcal{M}^1_{loc}$, i.e. they are all locally finite (Radon) measures. If $n=1$, the situation is well-known, and $BV_{loc}\subset L^\infty_{loc}$. So assume $n\geq 2$. Since $W^{s,p}_{c}(\mathbb{R}^n)\subset C^0_{c}(\mathbb{R}^n)$ if $s>n/p$, you have that $$BV_{loc}(\mathbb{R}^n)\subset W^{1-s,p'}_{loc}(\mathbb{R}^n)\subset L^{p'}_{loc}(\mathbb{R}^n)$$ if $s\leq 1$ and $1/p+1/p'=1$, that is if $p'<n/(n-1)$. On the other sidehand, when $n>1$, $1/r^\alpha$ is in $BV_{loc}(\mathbb{R}^n)$ if $\alpha<n-1$, since partial derivatives are in $L^1_{loc}$, but it is in $L^q$ only for $q<n/\alpha$, so that $BV_{loc}(\mathbb{R}^n)\subset L^q_{loc}(\mathbb{R}^n)$ fails for any $q>n/(n-1)$. I wouldn't bet on the limiting case.