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We will give new upper bounds for the number of solutions to the inequalities of the shape \vert F(x,y)\vert \leq h, where F(x,y) is a sparse binary form, with integer coefficients, and h is a sufficiently small integer in terms of the discriminant of the binary form F. Our bounds depend on the number of non-vanishing coefficients of F(x,y). When F is ''really sparse'', we establish a sharp upper bound for the number of solutions that is linear in terms of the number of non-vanishing coefficients. This work will provide affirmative answers to a number of conjectures posed by Mueller and Schmidt in [Trans. Amer. Math. Soc. 303 (1987), pp. 241-255], [Acta Math. 160 (1988), pp. 207-247], in special but important cases. [ABSTRACT FROM AUTHOR]