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To solve the problem, we need to analyze the function \( f(x) = \frac{\ln |x|}{|x-1|} \sin x \) and identify the types of discontinuities it possesses.

### Step-by-Step Analysis:

1. **Identify Points of Discontinuity**:
   - \( x = 0 \) because \( \ln |x| \) is undefined at \( x = 0 \).
   - \( x = 1 \) because \( |x-1| \) results in division by zero at \( x = 1 \).

2. **Behavior at \( x = 0 \)**:
   - As \( x \to 0 \), \( \ln |x| \to -\infty \).
   - \( \sin x \) oscillates between -1 and 1, which doesn't affect the nature of the discontinuity since \(\ln |x|\) dominates.
   - This suggests an essential discontinuity at \( x = 0 \).

3. **Behavior at \( x = 1 \)**:
   - As \( x \to 1 \), \( |x-1| \to 0 \), causing the denominator to approach zero.
   - The numerator \( \ln |x| \sin x \) does not approach zero in a similar manner to counteract the zero in the denominator.
   - This suggests an essential discontinuity at \( x = 1 \).

### Conclusion:

The function \( f(x) = \frac{\ln |x|}{|x-1|} \sin x \) has essential discontinuities at both \( x = 0 \) and \( x = 1 \). There are no removable or jump discontinuities as both points involve division by zero or logarithmic singularities which cannot be canceled out.

### Answer:

The correct answer is (D) \(2\) 个无穷间断点 (2 essential discontinuities).

Do you have any further questions or need more details on any step? Here are some related questions you might find useful:

1. What is an essential discontinuity, and how is it different from a removable or jump discontinuity?
2. How does the sine function impact the behavior of the function near points of discontinuity?
3. Can you provide more examples of functions with essential discontinuities?
4. How do logarithmic functions behave near their points of discontinuity?
5. What are some methods to identify discontinuities in a complex function?

**Tip:** When dealing with functions involving absolute values, logarithms, or trigonometric functions, always consider their behavior at critical points where the argument of the function might cause it to be undefined or approach infinity.

Learn how to analyze the types of discontinuities in the function f(x) = ln |x| / |x-1| sin x. This problem involves understanding the behavior of logarithmic, absolute value, and trigonometric functions at points of discontinuity. Suitable for advanced calculus students.

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