This choice mainly depends on what your output represents. Given a vector $\mathbf{x}$, the sigmoid function is given by
$$ \sigma(x_i) = \frac{\exp(x_i)}{1 + \exp(x_i)} $$
while the softmax is given by
$$ \mathrm{softmax}(x_i) = \frac{\exp(x_i)}{\sum_{j=0}^N \exp(x_j)}$$
A key difference is that the output of the sigmoid function applied to $x_i$ only depends on $x_i$. The output of the softmax function depends on all elements of the vector $\mathbf{x}$.
The sigmoid will squash each $x_i$ into the range $(0, 1)$, which enables you to interpret $\sigma(x_i)$ as the probability of $x_i$.
But, if you have multiple classes, e.g. 0-9 in MNIST, each probability is independent, that means you could have probability $p=0.9$ for the digit 1 and $p=0.5$ for the digit 3. This is undesirable if you want to distinguish between multiple classes. However, if multiple classes can appear at the same time, then sigmoid is well suited.
Now, the softmax is basically a sigmoid function which is normalized such that $\sum_{j=0}^N \mathrm{softmax}(x_j) = 1$. This means that the output of a softmax layer is a valid probability mass function, i.e. the output of your neural network is the probability of the input belonging to a certain class.
So, if you want to classify between e.g. the 10 digits as in MNIST, then softmax is the way to go. If the classes are independent, i.e. it is possible that an image shows a cat and a dog at the same time, then sigmoid is the way to go.
