The simple answer is yes, it is permitted to tie shoelaces on Shabbat.

But just in case you are interested, allow me to share a little background information:

Altogether there are 39 general areas of creative work that are forbidden on Shabbat. According to the Oral Law, this is the amount of creative acts that were required for the construction effort of the Mishkan (Tabernacle) in the desert. (See The 39 Melachot for a complete listing of these categories of work.) A peculiar arrangement of certain verses in the Book of Exodus teaches us that those activities required to construct the Mishkan are those that we must desist from doing during our weekly day of rest.

One of the 39 categories of forbidden activities is tying a knot. Knotting was needed in the construction of the Mishkan for two reasons: a) Threads were knotted in the process of creating the curtains and tapestries that covered the Mishkan and hung in its doorways. b) Certain animals whose byproducts were used for the construction were trapped in knotted snares.1

Nevertheless, it is only forbidden to create a permanent, or "long term," knot, similar to the Mishkan knots. But what constitutes a permanent knot?

There are those who see this purely as a time issue, and therefore opine than any knot that normally doesn't last longer than a day (like shoelaces) is permitted.

Others are of the opinion that any "professional knot" (like the ones which are taught to boy scouts) is considered permanent and should therefore not be tied on Shabbat regardless of how long it normally lasts. Now, while what exactly constitutes a "professional" knot may be up for debate, a simple bow know certainly does not fall in to this category.

In practice we follow both opinions and therefore don't make any knots which will last for extended periods of time nor do we make any which can be considered "professional."

The bow knot on a shoe is normally untied at night, when the shoes are removed. It is therefore no problem to tie and untie shoes on Shabbat.

Yours truly,

Rabbi Menachem Posner