The choice of which molecule to keep constant and which to vary will depend on the experiment type, the properties of the molecules and the stoichiometry of the interaction.
For ITC, where neither protein is labeled or immobilized, either molecule can be in the cell or syringe. You will need a higher concentration and total amount of the molecule in the syringe, so solubility and availability will generally be deciding factors. For protein/small molecule interactions, it is most common to put the protein in the cell and the compound in the syringe.
For MST, one of the molecules must be fluorescently labeled. You'll first start by determining which molecules can be easily labeled with a fluorophore. If both molecules can be labeled (as with protein/protein interactions), your choice will generally be based on solubility and availability. You will use a constant low concentration (typically nM) of the fluorescent molecule and much higher concentration of the non-fluorescent binding partner (100X Kd). For protein/small molecule interactions, it is most common to fluorescently label the protein and have a variable concentration of non-fluorescent compound.
For SPR and BLI, one molecule must be tethered to the biolayer surface. There are variety of ways to capture proteins onto SPR and BLI sensors, including covalent capture and many affinity capture options. For protein/protein interactions you could immobilize either molecule. However, it's imporant to consider stoichiometry of binding. If one of the molecules is multivalent (an antibody, for example), then you should always try to immobilize the multivalent molecule, as the fitting is much more straightforward. Other considerations are solubility, availability and existing tags for immobilization. For protein/small molecule interactions, it is most common to immobilize the protein and use a variable concentration of compound.