Here's a way to get the local rotation of just the y-axis. This function can be modified to get the x or z-axis.

/// <summary> isolate the y-Component of a rotation </summary>
private Quaternion yRotation(Quaternion q)
{
    float theta = Mathf.Atan2(q.y, q.w);

    // quaternion representing rotation about the y axis
    return new Quaternion(0, Mathf.Sin(theta), 0, Mathf.Cos(theta));
}

You can verify the result in the Unity inspector by converting to Euler:

public float yLocal;
void Update()
{
    yLocal = yRotation(this.transform.rotation).eulerAngles.y;
}

EDIT: The below quickly breaks down when more than one euler axis is non-zero. Refer to Component of a quaternion rotation around an axis instead.

(thank you @LyrePyre for noticing and @minorlogic for the link).

Original:

Here's a way to get the local rotation of just the y-axis. This function can be modified to get the x or z-axis.

/// <summary> isolate the y-Component of a rotation </summary>
private Quaternion yRotation(Quaternion q)
{
    float theta = Mathf.Atan2(q.y, q.w);

    // quaternion representing rotation about the y axis
    return new Quaternion(0, Mathf.Sin(theta), 0, Mathf.Cos(theta));
}

public float yLocal;
void Update()
{
    yLocal = yRotation(this.transform.rotation).eulerAngles.y;
}

Here's a way to get the local rotation of just the y-axis. This function can be modified to get the x or z-axis.

/// <summary> isolate the y-Component of a rotation </summary>
private Quaternion yRotation(Quaternion q)
{
    float theta = Mathf.Atan2(q.y, q.w);

    // quaternion representing rotation about the y axis
    return new Quaternion(0, Mathf.Sin(theta), 0, Mathf.Cos(theta));
}

Here's a way to get the local rotation of just the y-axis. This function can be modified to get the x or z-axis.

/// <summary> isolate the y-Component of a rotation </summary>
private Quaternion yRotation(Quaternion q)
{
    float theta = Mathf.Atan2(q.y, q.w);

    // quaternion representing rotation about the y axis
    return new Quaternion(0, Mathf.Sin(theta), 0, Mathf.Cos(theta));
}

You can verify the result in the Unity inspector by converting to Euler:

public float yLocal;
void Update()
{
    yLocal = yRotation(this.transform.rotation).eulerAngles.y;
}