Second attempt - probably just as bad as my first, code is neater though.

Levenshtein.Intrinsics.cs

#if NETCOREAPP3_0
using System;
using System.Buffers;
using System.Runtime.CompilerServices;
using System.Runtime.Intrinsics.X86;
using System.Runtime.Intrinsics;

namespace Quickenshtein
{
	/// <summary>
	/// Quick Levenshtein Distance Calculator
	/// </summary>
	public static partial class Levenshtein
	{
		[MethodImpl(MethodImplOptions.AggressiveInlining)]
		private static unsafe int CalculateDistance_Intrinsics(ReadOnlySpan<char> source, ReadOnlySpan<char> target)
		{
			var sourceLength = source.Length;
			var targetLength = target.Length;

			var arrayPool = ArrayPool<ushort>.Shared;
			var pooledArray = arrayPool.Rent(targetLength);
			Span<ushort> previousRow = pooledArray;

			//ArrayPool values are sometimes bigger than allocated, let's trim our span to exactly what we use
			previousRow = previousRow.Slice(0, targetLength);

			FillRow(previousRow);

			var negationVector = Vector128.Create((ushort)1);
			var allOnesVector128 = negationVector;

			fixed (char* targetPtr = target)
			fixed (ushort* previousRowPtr = previousRow)
			{
				var targetUShortPtr = (ushort*)targetPtr;

				for (ushort rowIndex = 0; rowIndex < sourceLength; rowIndex++)
				{
					var lastSubstitutionCost = rowIndex;
					var lastInsertionCost = (ushort)(rowIndex + 1);

					var sourcePrevChar = source[rowIndex];

					var columnIndex = 0;
					ushort lastDeletionCost;
					ushort localCost;

					var rowColumnsRemaining = targetLength;
					var sourcePrevCharVector = Vector128.Create(sourcePrevChar);

					if (rowColumnsRemaining >= 8)
					{
						var substitutionCostsVector = Vector128.Create(
							lastSubstitutionCost, 
							previousRowPtr[0],
							previousRowPtr[1],
							previousRowPtr[2],
							previousRowPtr[3],
							previousRowPtr[4],
							previousRowPtr[5],
							previousRowPtr[6]);

						while (rowColumnsRemaining >= 8)
						{
							rowColumnsRemaining -= 8;

							var targetCharVector = Sse3.LoadDquVector128(targetUShortPtr + columnIndex);
							var notEqualVector = Sse2.AndNot(Sse2.CompareEqual(sourcePrevCharVector, targetCharVector), negationVector);

							var deletionCostsVector = Sse3.LoadDquVector128(previousRowPtr + columnIndex);
							var adjustedDeletionCostsVector = Sse2.Add(deletionCostsVector, allOnesVector128);

							var adjustedSubstitutionCostsVector = Sse2.Add(notEqualVector, substitutionCostsVector);

							lastInsertionCost = Sse41.MinHorizontal(Vector128.Create(
								(ushort)(lastInsertionCost + 1),
								adjustedDeletionCostsVector.GetElement(0),
								adjustedSubstitutionCostsVector.GetElement(0),
								ushort.MaxValue, ushort.MaxValue, ushort.MaxValue, ushort.MaxValue, ushort.MaxValue
							)).GetElement(0);
							previousRowPtr[columnIndex++] = lastInsertionCost;

							lastInsertionCost = Sse41.MinHorizontal(Vector128.Create(
								(ushort)(lastInsertionCost + 1),
								adjustedDeletionCostsVector.GetElement(1),
								adjustedSubstitutionCostsVector.GetElement(1),
								ushort.MaxValue, ushort.MaxValue, ushort.MaxValue, ushort.MaxValue, ushort.MaxValue
							)).GetElement(0);
							previousRowPtr[columnIndex++] = lastInsertionCost;

							lastInsertionCost = Sse41.MinHorizontal(Vector128.Create(
								(ushort)(lastInsertionCost + 1),
								adjustedDeletionCostsVector.GetElement(2),
								adjustedSubstitutionCostsVector.GetElement(2),
								ushort.MaxValue, ushort.MaxValue, ushort.MaxValue, ushort.MaxValue, ushort.MaxValue
							)).GetElement(0);
							previousRowPtr[columnIndex++] = lastInsertionCost;

							lastInsertionCost = Sse41.MinHorizontal(Vector128.Create(
								(ushort)(lastInsertionCost + 1),
								adjustedDeletionCostsVector.GetElement(3),
								adjustedSubstitutionCostsVector.GetElement(3),
								ushort.MaxValue, ushort.MaxValue, ushort.MaxValue, ushort.MaxValue, ushort.MaxValue
							)).GetElement(0);
							previousRowPtr[columnIndex++] = lastInsertionCost;

							lastInsertionCost = Sse41.MinHorizontal(Vector128.Create(
								(ushort)(lastInsertionCost + 1),
								adjustedDeletionCostsVector.GetElement(4),
								adjustedSubstitutionCostsVector.GetElement(4),
								ushort.MaxValue, ushort.MaxValue, ushort.MaxValue, ushort.MaxValue, ushort.MaxValue
							)).GetElement(0);
							previousRowPtr[columnIndex++] = lastInsertionCost;

							lastInsertionCost = Sse41.MinHorizontal(Vector128.Create(
								(ushort)(lastInsertionCost + 1),
								adjustedDeletionCostsVector.GetElement(5),
								adjustedSubstitutionCostsVector.GetElement(5),
								ushort.MaxValue, ushort.MaxValue, ushort.MaxValue, ushort.MaxValue, ushort.MaxValue
							)).GetElement(0);
							previousRowPtr[columnIndex++] = lastInsertionCost;

							lastInsertionCost = Sse41.MinHorizontal(Vector128.Create(
								(ushort)(lastInsertionCost + 1),
								adjustedDeletionCostsVector.GetElement(6),
								adjustedSubstitutionCostsVector.GetElement(6),
								ushort.MaxValue, ushort.MaxValue, ushort.MaxValue, ushort.MaxValue, ushort.MaxValue
							)).GetElement(0);
							previousRowPtr[columnIndex++] = lastInsertionCost;

							lastInsertionCost = Sse41.MinHorizontal(Vector128.Create(
								(ushort)(lastInsertionCost + 1),
								adjustedDeletionCostsVector.GetElement(7),
								adjustedSubstitutionCostsVector.GetElement(7),
								ushort.MaxValue, ushort.MaxValue, ushort.MaxValue, ushort.MaxValue, ushort.MaxValue
							)).GetElement(0);
							previousRowPtr[columnIndex++] = lastInsertionCost;

							lastSubstitutionCost = deletionCostsVector.GetElement(7);

							if (rowColumnsRemaining >= 8)
							{
								substitutionCostsVector = Sse3.LoadDquVector128(previousRowPtr + columnIndex - 1)
									.WithElement(0, lastSubstitutionCost);
							}
						}
					}

					if (rowColumnsRemaining > 4)
					{
						rowColumnsRemaining -= 4;

						localCost = lastSubstitutionCost;
						lastDeletionCost = previousRowPtr[columnIndex];
						if (sourcePrevChar != targetPtr[columnIndex])
						{
							localCost = Math.Min(lastInsertionCost, localCost);
							localCost = Math.Min(lastDeletionCost, localCost);
							localCost++;
						}
						lastInsertionCost = localCost;
						previousRowPtr[columnIndex++] = localCost;
						lastSubstitutionCost = lastDeletionCost;

						localCost = lastSubstitutionCost;
						lastDeletionCost = previousRowPtr[columnIndex];
						if (sourcePrevChar != targetPtr[columnIndex])
						{
							localCost = Math.Min(lastInsertionCost, localCost);
							localCost = Math.Min(lastDeletionCost, localCost);
							localCost++;
						}
						lastInsertionCost = localCost;
						previousRowPtr[columnIndex++] = localCost;
						lastSubstitutionCost = lastDeletionCost;

						localCost = lastSubstitutionCost;
						lastDeletionCost = previousRowPtr[columnIndex];
						if (sourcePrevChar != targetPtr[columnIndex])
						{
							localCost = Math.Min(lastInsertionCost, localCost);
							localCost = Math.Min(lastDeletionCost, localCost);
							localCost++;
						}
						lastInsertionCost = localCost;
						previousRowPtr[columnIndex++] = localCost;
						lastSubstitutionCost = lastDeletionCost;

						localCost = lastSubstitutionCost;
						lastDeletionCost = previousRowPtr[columnIndex];
						if (sourcePrevChar != targetPtr[columnIndex])
						{
							localCost = Math.Min(lastInsertionCost, localCost);
							localCost = Math.Min(lastDeletionCost, localCost);
							localCost++;
						}
						lastInsertionCost = localCost;
						previousRowPtr[columnIndex++] = localCost;
						lastSubstitutionCost = lastDeletionCost;
					}

					while (rowColumnsRemaining > 0)
					{
						rowColumnsRemaining--;

						localCost = lastSubstitutionCost;
						lastDeletionCost = previousRowPtr[columnIndex];
						if (sourcePrevChar != targetPtr[columnIndex])
						{
							localCost = Math.Min(lastInsertionCost, localCost);
							localCost = Math.Min(lastDeletionCost, localCost);
							localCost++;
						}
						lastInsertionCost = localCost;
						previousRowPtr[columnIndex++] = localCost;
						lastSubstitutionCost = lastDeletionCost;
					}
				}
			}

			var result = previousRow[targetLength - 1];
			arrayPool.Return(pooledArray);
			return result;
		}
	}
}
#endif