Count trailing truths

Dyalog APL, 6 2 bytes

⊥⍨

Test it on TryAPL.

How it works

⊥ (uptack, dyadic: decode) performs base conversion. If the left operand is a vector, it performs mixed base conversion, which is perfect for this task.

For a base vector b = b_n, ⋯, b₀ and a digit vector a = a_n, ⋯, a₀, b ⊥ a converts a to the mixed base b, i.e., it computes b₀⋯b_n-1a_n + ⋯ + b₀b₁a₂ + b₀a₁ + a₀.

Now, ⍨ (tilde dieresis, commute) modifies the operator to the left as follows. In a monadic context, it calls the operator with equal left and right arguments.

For example, ⊥⍨ a is defined as a ⊥ a, which computes a₀⋯a_n + ⋯ + a₀a₁a₂ + a₀a₁ + a₀, the sum of all cumulative products from the right to the left.

For k trailing ones, the k rightmost products are 1 and all others are 0, so their sum is equal to k.

JavaScript (ES6), 21 bytes

f=l=>l.pop()?f(l)+1:0

Test cases

f=l=>l.pop()?f(l)+1:0

console.log(f([])); // → 0
console.log(f([0])); // → 0
console.log(f([1])); // → 1
console.log(f([0, 1, 1, 0, 0])); // → 0
console.log(f([1, 1, 1, 0, 1])); // → 1
console.log(f([1, 1, 0, 1, 1])); // → 2
console.log(f([0, 0, 1, 1, 1])); // → 3
console.log(f([1, 1, 1, 1, 1, 1])); // → 6

Jelly, 4 bytes

ŒrṪP

Try it online! or Verify all test cases.

For the case where the list is empty, there are some curious observations. First, run-length encoding the empty list [] returns another empty list []. Then retreiving the last element from that using tail Ṫ returns 0 instead of a pair [value, count] which are the regular elements of a run-length encoded array. Then product P returns 0 when called on 0 which is the expected result.

Explanation

ŒrṪP  Main link. Input: list M
Œr    Run-length encode
  Ṫ   Tail, get the last value
   P  Product, multiply the values together