In this paper we study the scaling relations for the triggering of the fast, or ideal, tearing instability starting from equilibrium configurations relevant to astrophysical as well as laboratory plasmas that differ from the simple Harris current sheet configuration. We present the linear tearing instability analysis for equilibrium magnetic fields which a) go to zero at the boundary of the domain and b) contain a double current sheet system (the latter previously studied as a cartesian proxy for the m=1 kink mode in cylindrical plasmas). More generally, we discuss the critical aspect ratio scalings at which the growth rates become independent of the Lundquist number $S$, in terms of the dependence of the $Delta$ parameter on the wavenumber $k$ of unstable modes. The scaling $Delta(k)$ with $k$ at small $k$ is found to categorize different equilibria broadly: the critical aspect ratios may be even smaller than $L/a sim S^{alpha}$ with $alpha=1/3$ originally found for the Harris current sheet, but there exists a general lower bound $alphage1/4$.
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