How to draw a signature and save it to disc as a Bitmap?

Never use [Control].CreateGraphics, unless you need this object for immediate use in a specific environment. To measure the size of text when drawn in a specific graphics context, for example.
When you need the drawing to persist, as painting on the surface of a control, use the Graphics object provided by the PaintEventArgs of the Paint event (or similar event, as the DrawItem event of the ComboBox, ListBox, ListView controls).
You’ll find this kind of recommendations all over SO (and the Web, in general).

How to proceed:

1. We need an object that can store the Mouse/Pen movements that define a hand-made curve.
2. The object needs to store a new curve definition each time the left button of the mouse is released (or the Pen touch is lost and re-acquired).
3. We need a Graphics object that can convert the cardinal spline curves, defined by the points created by Mouse/Pen movements, to Bezier curves (combinations of curves – outlines that represents a series of smooth vector lines – are often referred to as Paths in vector graphics).

Here, the object that stores the movements is a Dictionary(Of Integer, List(Of Point)), where the Key represents a curve and the Value represents the collection of Points that define that curve.
Each time the Left Button of the Mouse is pressed, a new Key is created and a new List(Of Point) is associated with the Key.
When the Mouse/Pen is moved, the new Point location is added to the current curve’s List(Of Point).

The GraphicsPath class can convert the List(Of Point) collection to Control Points of a Bezier curve, using the GraphicsPath.AddCurve() method.
This method accepts an array of Points and a Tension value as arguments. The Tension is a value between 0 and 1 that defines the amount of bending applied to a curve when connecting Points. A value of 0.5f is used here.

► When we need to paint the drawing on a Bitmap to save the results to disc, we apply the same logic to the Graphics object derived, in this case, from a Bitmap object.
Thus, just one method is used to draw both on the surface of a Control and onto a Bitmap object.
The DrawSignature(g As Graphics) method in this code.

This is how it works:

Code that reproduces the procedure described:

pBoxSignature is the PictureBox where the signature is drawn
pBoxSavedSignature is the (smaller) PictureBox used to show the last image saved
btnClearSignature is the Button used to clear the signature
btnSaveSignature is the Button used to save the signature to a new Image
txtSignatureFileName is the TextBox used to enter the signature’s file name

Private signatureObject As New Dictionary(Of Integer, List(Of Point))
Private signaturePen As New Pen(Color.Black, 4)
Private currentCurvePoints As List(Of Point)
Private currentCurve As Integer = -1

Private Sub pBoxSignature_MouseDown(sender As Object, e As MouseEventArgs) Handles pBoxSignature.MouseDown
    currentCurvePoints = New List(Of Point)
    currentCurve += 1
    signatureObject.Add(currentCurve, currentCurvePoints)
End Sub

Private Sub pBoxSignature_MouseMove(sender As Object, e As MouseEventArgs) Handles pBoxSignature.MouseMove
    If e.Button <> MouseButtons.Left OrElse currentCurve < 0 Then Return
    signatureObject(currentCurve).Add(e.Location)
    pBoxSignature.Invalidate()
End Sub

Private Sub btnClearSignature_Click(sender As Object, e As EventArgs) Handles btnClearSignature.Click
    currentCurve = -1
    signatureObject.Clear()
    pBoxSignature.Invalidate()
End Sub

Private Sub btnSaveSignature_Click(sender As Object, e As EventArgs) Handles btnSaveSignature.Click
    Dim signatureFileName = txtSignatureFileName.Text.Trim()
    If String.IsNullOrEmpty(signatureFileName) Then Return
    If currentCurve < 0 OrElse signatureObject(currentCurve).Count = 0 Then Return

    imgSignature As Bitmap = New Bitmap(pBoxSignature.Width, pBoxSignature.Height, PixelFormat.Format32bppArgb) 
    Using g As Graphics = Graphics.FromImage(imgSignature)
        DrawSignature(g)

        Dim signaturePath As String = Path.Combine(Application.StartupPath, $"{signatureFileName}.png")
        imgSignature.Save(signaturePath, ImageFormat.Png)
        pBoxSavedSignature.Image?.Dispose()
        pBoxSavedSignature.Image = imgSignature
    End Using
End Sub

Private Sub pBoxSignature_Paint(sender As Object, e As PaintEventArgs) Handles pBoxSignature.Paint
    If currentCurve < 0 OrElse signatureObject(currentCurve).Count = 0 Then Return
    DrawSignature(e.Graphics)
End Sub

Private Sub DrawSignature(g As Graphics)
    g.CompositingQuality = CompositingQuality.HighQuality
    g.SmoothingMode = SmoothingMode.AntiAlias

    For Each curve In signatureObject
        If curve.Value.Count < 2 Then Continue For
        Using gPath As New GraphicsPath()
            gPath.AddCurve(curve.Value.ToArray(), 0.5F)
            g.DrawPath(signaturePen, gPath)
        End Using
    Next
End Sub

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C# Version

private Dictionary<int, List<Point>> signatureObject = new Dictionary<int, List<Point>>();
private Pen signaturePen = new Pen(Color.Black, 4);
private List<Point> currentCurvePoints = null;
private int currentCurve = -1;

private void pBoxSignature_MouseDown(object sender, MouseEventArgs e)
{
    currentCurvePoints = new List<Point>();
    currentCurve += 1;
    signatureObject.Add(currentCurve, currentCurvePoints);
}

private void pBoxSignature_MouseMove(object sender, MouseEventArgs e)
{
    if (e.Button != MouseButtons.Left || currentCurve < 0) return;
    signatureObject[currentCurve].Add(e.Location);
    pBoxSignature.Invalidate();
}

private void btnClearSignature_Click(object sender, EventArgs e)
{
    currentCurve = -1;
    signatureObject.Clear();
    pBoxSignature.Invalidate();
}

private void btnSaveSignature_Click(object sender, EventArgs e)
{
    string signatureFileName = txtSignatureFileName.Text.Trim();
    if (string.IsNullOrEmpty(signatureFileName)) return;
    if (currentCurve < 0 || signatureObject[currentCurve].Count == 0) return;

    var imgSignature = new Bitmap(pBoxSignature.Width, pBoxSignature.Height, PixelFormat.Format32bppArgb);
    using (var g = Graphics.FromImage(imgSignature)) {
        DrawSignature(g);
        string signaturePath = Path.Combine(Application.StartupPath, $"{signatureFileName}.png");
        imgSignature.Save(signaturePath, ImageFormat.Png);
        pBoxSavedSignature.Image?.Dispose();
        pBoxSavedSignature.Image = imgSignature;
    }
}

private void pBoxSignature_Paint(object sender, PaintEventArgs e)
{
    if (currentCurve < 0 || signatureObject[currentCurve].Count == 0) return;
    DrawSignature(e.Graphics);
}

private void DrawSignature(Graphics g)
{
    // In case you have composited background
    g.CompositingQuality = CompositingQuality.HighQuality;
    g.SmoothingMode = SmoothingMode.AntiAlias;

    foreach (var curve in signatureObject) {
        if (curve.Value.Count < 2) continue;

        using (var gPath = new GraphicsPath()) {
            gPath.AddCurve(curve.Value.ToArray(), 0.5f);
            g.DrawPath(signaturePen, gPath);
        }
    }
}