For stranded copper, the best connector depends on the job, but in most everyday wiring situations a properly sized twist-on wire connector rated for stranded wire or a good lever-style connector is a solid choice. If you want the most reliable mechanical connection, many electricians prefer connectors that are specifically listed for stranded conductors rather than ones meant only for solid wire. The key is not just the brand or style, but whether the connector is approved for the wire size and wire type you’re using.
For small splices in a junction box, lever connectors often work very well with stranded copper because they clamp the conductor firmly and make it easy to verify that the wire is fully seated. They’re especially handy when you’re joining stranded to stranded, or stranded to solid, as long as the connector rating allows it. Traditional twist-on wire nuts can also work fine with stranded copper, but they need the right size and a clean installation. If the nut is too large, the wires can back out. If it’s too small, you can damage the strands or get an incomplete connection.
If you’re dealing with fixture leads, appliance repairs, or panel-related terminations, crimp connectors or ferrules may be the better option. Ferrules are especially useful when stranded copper is landing under a screw terminal, because they keep the strands together and reduce the chance of loose strands spreading out. That said, ferrules only help if the terminal is designed to accept them and you crimp them with the correct tool. A poorly crimped ferrule is worse than no ferrule at all.
For high-vibration or exposed connections, a properly crimped, insulated butt splice or ring terminal can be more dependable than a simple splice connector, again assuming it matches the wire gauge and application. In automotive, marine, and equipment wiring, crimp terminals are often preferred because they handle movement better than twist connections. In house wiring, though, you usually want a listed splice connector inside an approved box.
What I would avoid is mixing connector types that are not rated for the conductor combination, or using a connector that feels tight but isn’t actually listed for stranded copper. Also avoid nicking or over-stripping the wire, since damaged strands can weaken the connection. If the wire keeps slipping, the problem is often connector size or installation, not stranded copper itself.