1. **State the problem:** Find the limit as $x \to 0$ of $$\frac{(\sqrt[4]{1 + x^2} - 1) \sin x}{e^x - 1}.$$\n\n2. **Recall formulas and rules:**\n- For small $x$, $\sin x \approx x$.\n- For small $x$, $e^x - 1 \approx x$.\n- Use binomial expansion for $\sqrt[4]{1 + x^2} = (1 + x^2)^{1/4}$.\n\n3. **Expand $\sqrt[4]{1 + x^2}$:**\nUsing binomial approximation for small $x$,\n$$ (1 + x^2)^{1/4} \approx 1 + \frac{1}{4}x^2. $$\nSo, $$ \sqrt[4]{1 + x^2} - 1 \approx \frac{1}{4}x^2. $$\n\n4. **Substitute approximations into the limit:**\n$$ \lim_{x \to 0} \frac{(\frac{1}{4}x^2)(x)}{x} = \lim_{x \to 0} \frac{\frac{1}{4}x^3}{x} = \lim_{x \to 0} \frac{1}{4}x^2. $$\n\n5. **Evaluate the limit:**\nAs $x \to 0$, $\frac{1}{4}x^2 \to 0$.\n\n**Final answer:** $$\boxed{0}.$$