What is the type signature of a Turing Machine?
Usually, Turing machines are presented using set theory rather than type theory.

There might be many ways to represent a TM in type theory, each one with its advantages and disadvantages. I would not represent it as a function $\{0,1\}^* \to\{0,1\}^*$ since the function implemented by a TM is a partial function, because the TM might fail to terminate, while in type theory functions are total.

If we wanted to follow the set theoretic definition, we might define a TM as a dependent tuple. This tuple would include:


$\Sigma:\sf Type$ a type representing the tape "alphabet"
$Q:\sf Type$ a type representing the states
$b:\Sigma$ the blank symbol
$q_0 : Q$ the initial state
$\delta : Q \to \Sigma \to (Q \times \Sigma \times 3)$ for the transition function (assuming a deterministic TM) where $3=1+1+1$ represents the directions
$F : Q \to 2$ with $2=1+1$ the halting states


Technically, this would be a huge dependent sum such as

$$
\sum_{\Sigma\sf Type}
\sum_{Q:\sf Type}
  \Sigma \times Q \times (Q\to \Sigma \to(Q\times \Sigma\times 3))\times(Q\to 2)
$$

The above would be a possible representation of a TM. (I'm unsure it would be a very good one, especially if we work without functional extensionality, but I'll neglect that.)

In a similar way one could define the type of the configurations of a TM ${\sf Conf}\ T$ and a step function $step: {\sf Conf}\ T \to 1 + {\sf Conf}\ T$
(the $1+$ is to report "no next step" when the TM halts).

One could even define a function to run $n$ steps, or a predicate for "the TM $T$ on input $x$ halts with output $y$".

$$
\begin{array}{l}
{\sf runTM} : 
\prod_{T:TM} {\sf Conf}\ T \to \mathbb N \to 1+{\sf Conf}\ T
\\
{\sf semanticsTM} : 
\prod_{T:TM} (\pi_{\Sigma} T)^* \to (\pi_{\Sigma} T)^* \to \sf Type
\end{array}
$$