Your question is How to bring rows with search results on top in the datagridview and your current approach is attempting to use DataGridView.Sort to work with the UI control directly.

If you're open to doing this another way, with DataGridView it's often easier to achieve your desired outcome by setting the DataSource property (in this case, a BindingList<Appliance>) and working with that list instead.

For starters, the DGV can be completely set up using ~10 lines in the OnLoad of the main form.

public partial class MainForm : Form
{
    public MainForm() => InitializeComponent();
    private BindingList<Appliance> Appliances { get; }  = new BindingList<Appliance>();
    protected override void OnLoad(EventArgs e)
    {
        base.OnLoad(e);

        // Set up the DGV and add a few appliances.
        dtgridappliance.DataSource = Appliances;
        dtgridappliance.AllowUserToAddRows = false;
        Appliances.Add(new Appliance(ApplianceType.Refrigerator, "Maytag" ));
        Appliances.Add(new Appliance(ApplianceType.Refrigerator, "LG" ));
        Appliances.Add(new Appliance(ApplianceType.Microwave, "Amana" ));
        Appliances.Add(new Appliance(ApplianceType.Dishwasher, "Samsung" ));
        Appliances.Add(new Appliance(ApplianceType.Dishwasher, "Whirlpool" ));
        dtgridappliance.AllowUserToAddRows = false;
        // Format columns
        dtgridappliance.Columns[nameof(Appliance.Name)].AutoSizeMode = DataGridViewAutoSizeColumnMode.Fill;

        // Populate the combo box and dd a handler for when the combo box changes
        object[] values = Enum.GetValues(typeof(ApplianceType)).Cast<object>().ToArray();
        comboBoxSearch.Items.AddRange(values);
        comboBoxSearch.SelectedIndex = (int)ApplianceType.Refrigerator;
        comboBoxSearch.SelectionChangeCommitted += onSearchByApplianceType;
    }
}

Where the class representing a Row of data looks something like this:

enum ApplianceType { Dishwasher, Microwave, Refrigerator  }
class Appliance
{
    public Appliance(ApplianceType applianceType, string name)
    {
        ApplianceType = applianceType;
        Name = name;
    }
    public ApplianceType ApplianceType { get; private set; }
    public string Name { get; private set; }
}

---

The sort requires a way to compare a given Appliance to the type selected in the combo box:

private int compareToSelectedType(Appliance appliance)
{
    return appliance.ApplianceType.Equals(comboBoxSearch.SelectedItem) ?
        0 : 1; // Ones that match will sort before ones that don''t.
}

---

Now, when the combo box changes, you can remove all the DataGridView items and put them back in a different order.

private void onSearchByApplianceType(object? sender, EventArgs e)
{
    // Use System.Linq to sort a new temporary list.
    var tmp = new List<Appliance>(
        Appliances
        .OrderBy(_=>compareToSelectedType(_))
        .ThenBy(_ => _.ApplianceType)
        .ThenBy(_=>_.Name));

    // Remove all items from the DataGridView
    Appliances.Clear();

    // Put them back in with the new order.
    foreach (var appliance in tmp)
    {
        Appliances.Add(appliance);
    }
}

This is "just one way" to do it. Hope this helps get you closer to your intended outcome.