There are typical transits under which you can meet a love interest, whether it's just a lasting fling or someone you will spend your life with. Remember that only a detailed study of your chart can give you sure and in-depth explanations; and the transits described below are potentialities.
As a rule of thumb (= not every single time for every single person, but very often) when meeting someone we find an outer planet aspecting:
Venus (a love interest)
The ruler of the 7th House or the Descendant cusp itself (ruler: Planet ruling the sign on the cusp of this House. Ex: Cancer is on the cusp of the 7th House, Moon is the ruler) (a relationship)
Sun (a man enters your life)
Moon (a woman enters your life OR new emotional bond)
Mars (a sexual interest, more so a man)
Note, I bypassed the question of sexuality by indicating what the planet represents. When Sun is aspected, it represents your sense of self being modified, + the possibility of a masculine figure entering your life. If you're heterosexual, this means a male partner. If you're a gay man, this can show a new partner. I made it simpler by indicating what the planet represents, choose whichever fits your sexuality / relationship desires. If the question of gender is unclear for you or your potential partner, you can make it even more simpler with the following guideline: Moon=domesticity, Mars=sex, Venus=love.
Jupiter transit Venus / Moon / ruler of 7th house or DS cusp / Sun / Mars
A lot of relationships start under Jupiter conjunct, sextile, or trine Venus. Venus is our attractions, and Jupiter is the planet of legalization. When Jupiter transits a planet, it reorganizes it, legalizes it, officializse it. Therefore it's not surprising that relationships appear under such a transit. If you're already in a relationship, this can be a time for actual legalization (common law, marriage).
Jupiter square or opposite Venus can be a time of break up, IF the relationship wasn't valid for you. It takes a few years for Jupiter to go from conjunction to square, so if you track your last relationships, chances are, you'll see the link between Jupiter and your beginnings / ends of relationship.
Jupiter in aspect to Sun is a time of reorganizing your identity and sense of self, and sometimes, it's through a new relationship, a masculine energy entering your life. Jupiter Moon can bring emotional satisfaction and abundance, through a new domestic relationship. Jupiter Mars often has more to do with work but can bring a sexual relationship. Energy is higher.
Saturn transit Venus / Moon / ruler of 7th house / Sun / Mars
Saturn conjunct, sextile, trine Venus is a great time to meet someone. The square and opposition can also represent an opportunity, especially for those who have strong Uranus or Neptune in their chart, as it "grounds" the person, but it can also represent delays or meeting someone who has a disconnected energy / lack of affection.
Often the person you meet is serious and committed, and is ready to jump into an "adult" relationship with you. If you're already in a relationship, such times will be extremely testing for you: The relationship will go through a series of test and survive, or either it won't survive and it will be over, or the relationship will strive and rise from the hardship. Don't beat yourself up over an incoming transit though, as one more time, Saturn does not break a relationship which would have survived life's hardship. Is it worth losing years over someone who won't cut it on the long run?
It is possible to meet someone under difficult transits (square, opposition), but you'll often find that the person you meet isn't fulfilling your needs, and will often trigger in you a need to develop self-love and boundaries.
Saturn square the AS can represent a time of commitment and doesn't necessarily break relationships, although it sometimes happens. Same remark for Saturn conjunct AS/DS.
Uranus transit Venus Moon / ruler of 7th house / DS / Sun / Mars
Uranus conjunct, sextile, trine Venus often coincides with a refreshing, exciting, unexpected and unpredictable relationship! Not necessarily for the long run, you need to find other transits to get an idea of the purpose of this relationship. All in all it is often a time where you free yourself from past patterns and discover new tastes, new passions for yourself. Perhaps you'll travel the world with this person? Or you'll adopt a relating style that you never had before!
Uranus conjunct, square and opposite will often coincide with the end of a relationship, because you feel the need to roll on your own. You may have abandoned some of your individuality for the sake of a partnership and it is a time during which you will reclaim your independence. Starting a relationship under such transit indicates that the relationship will often serve this purpose. It can also represent staying in the relationship but entirely changing how the relationship is organized, structured.
Uranus in aspect to Sun and Mars would be much more relevant to a man entering your life, Uranus-Sun typically brings a relationship with a man who is seen as an ideal, while the aspect to Mars is more so an attraction to someone who feels irresistible.
Uranus Moon is much more relevant for a woman entering your life but it can also stand for a new emotional development through a relationship that quickly bonds emotionally.
Surprisingly, Uranus square the AS/DS does sometimes bring a new relationship, while the trine is often more supportive. Uranus conjunct AS/DS is as well the possibility of a brand new relationship.
Pluto transit Venus / Moon / ruler of 7th house / Sun / Mars
Pluto conjunct, sextile, trine Venus can bring a passionate love affair in your life, although it's often more about compulsion than about love. We do have possibilities for trauma bonds here, so do be careful. Once again, depends on the chart and rulers as well as other transits happening at the time. Meeting someone under such transit can be extremely deep, passionate and obsessive.
Pluto square and opposite Venus can also represents meeting someone, although the love affair will often bring a fair share of pain and drama that will force your to re-examine where are your boundaries and how you feel about yourself, you only attract like-minded vibes and this transit may be a difficult reminder of it.
Pluto aspects to Sun are about the identity but in the chart of a woman it's often projected on a male partner. Pluto aspects to Moon, Sun and Mars are more so personal and talk about a need to change, transform, what's inadequate about the expression of these planets, but sometimes it does bring a soul transforming relationship which serves as vector for transformation. You'll find similar aspects in synastry and composite of this relationship, every time.
Neptune transit Venus / Moon / ruler of 7th house /Sun / Mars
Neptune conjunct, sextile, trine Venus will bring a very romantic encounter, and although it won't necessary be extremely realistic, you can meet someone that makes you feel like you're living a fairy tail. Be aware that once the transit is gone the fairy tail may end too, so be sure to have your feet on the ground or a close friend who assures this grounded perspective for you.
Neptune square and opposite Venus often represents the dissolution of a relationship, and meeting someone under such transit often indicates that the entire relationship will be lived for the purpose of ending it. Secret affairs, or dysfunctional relationships that you know you should end but still pursue because butterflies and pink goggles. The relationship will serve the purpose of knowing better in the future. Sometimes though, it shows up as a blissful relationship which, although will change significantly in the future, can be positive and healthy.
Neptune in aspect to Sun and Mars is very often found in the chart of someone who meet a man they find really alluring and mysterious. It's a time of feeling really compelled by someone. When Neptune aspects Moon, a woman or an emotional bond enters your life that is really compelling, unconditionally loving, forgiving and sweet. It can be an illusion especially if the aspect is hard. The only thing I would particularly warn against is when Neptune forms a conjunction, square or opposition to Mars, this is usually a bad omen and I would triple check who you're dealing with.
Neptune relationships often have a dependency component to it, but they can also be blissful and sweet.