In Romans 15:30, Paul when urging his readers to pray for him mentions a connection with the love of the Spirit as part of the basis for why they should intercede for him. This shows that an activity of the Spirit's love is to stimulate mutual intercession between believers.
Moreover, such loving activity was linked to specific requests that Paul details in the context. Paul was sure that his requests were pleasing to God and therefore he could expect a response of love from the Spirit when fellow Christians prayed for him. So when our specific petitions for other Christians are answered, we experience the love of the Spirit.
Of course, answering prayer is not the only way by which the Spirit displays love towards his people. The fact is that everything he does for them is an expression of his love. It was an act of love when he regenerated them at the onset of their new life as believers. As he leads them throughout the years of discipleship, all he does reveals his love.
When he brings a verse of Scripture into our minds, he is showing his love for us. In a church service, when a preacher is used by him to instruct us about Jesus, that experience is one of love from the Spirit because he delights to bring Jesus before our minds and hearts.
Paul reminds us that one of the Spirit's activities is to lead believers to address God as Father (Gal. 4:6). This cry is said by Paul to be a strong one, and one reason why it is strong is because it is a cry of love. Because of the work of the Spirit, we love God with all our hearts.
The love of the Spirit lets us appreciate why he is grieved when his people sin (Eph. 4:30). Grieving is usually a response of love that disapproves of what the object of love is doing or saying or thinking. The loving Spirit knows everything about the hearts of his people and responds to them. Because he is the Spirit of love, he will work to restore them spiritually.
We are used to thinking about the love of the Father and the love of the Son. And we should also think about the love of the Spirit.