How to apply indenting serialization only to some properties?

One possibility would be to write a custom Json converter for the specific types you need special handling and switch the formatting for them:

class Program
{
    static void Main()
    {
        var root = new Root
        {
            Array = new[] { "element 1", "element 2", "element 3" },
            Object = new Obj
            {
                Property1 = "value1",
                Property2 = "value2",
            },
        };
        var settings = new JsonSerializerSettings
        {
            Formatting = Formatting.Indented,
        };
        settings.Converters.Add(new MyConverter());

        string json = JsonConvert.SerializeObject(root, settings);
        Console.WriteLine(json);
    }
}

public class Root
{
    public string[] Array { get; set; }
    public Obj Object { get; set; }
}

public class Obj
{
    public string Property1 { get; set; }
    public string Property2 { get; set; }
}

class MyConverter : JsonConverter
{
    public override bool CanConvert(Type objectType)
    {
        return objectType == typeof(string[]) || objectType == typeof(Obj);
    }

    public override object ReadJson(JsonReader reader, Type objectType, object existingValue, JsonSerializer serializer)
    {
        throw new NotImplementedException();
    }

    public override void WriteJson(JsonWriter writer, object value, JsonSerializer serializer)
    {
        writer.WriteRawValue(JsonConvert.SerializeObject(value, Formatting.None));
    }
}

This will output:

{
  "Array": ["element 1","element 2","element 3"],
  "Object": {"Property1":"value1","Property2":"value2"}
}