Most anomalies have tiers. Higher tiers are more dangerous and pay more. A running anomaly shows an icon and a countdown on the Galaxy Map and in a system's System Details panel.
Adaptive Hulls affects up to 20 systems and rotates every hour. Hostiles and Armadas deal +100% damage at Tier 1, rising to +600% at Tier 4. With a G6+ ship, your positive FKR reputation gains rise from +500% at Tier 1 to +5,000% at Tier 4. Disrupted Energy Shields affects up to 10 systems and rotates every 30 minutes. Each combat round, Hostiles and Armadas have a 10% chance (Tier 1) to 35% chance (Tier 4) to bypass your shields. Its reputation bonus is the biggest: +800% at Tier 1 to +10,000% at Tier 4. Ripple of Violence has no drawback. It affects 1 system at a time and moves every 5 minutes. In that system, your Isolytic Cascade Damage against Hostiles goes up 400%.
Technological Advancement only runs in Assimilated space, from :00 to :20 and from :30 to :50 each hour. Hostiles and Armadas deal +20% to +40% more damage (Tiers 1 to 3). Anti-Barrier Innovations adds +90% to +250% Apex Shred. Assimilation gives them a 100% chance each combat round to apply Assimilated to your ship for 1 round. The patch notes list no reputation bonus. They say to join the special events that run during this anomaly for rewards.
The Update 93 patch notes list a rotation calendar from 23 July to 18 August. Scopely says more anomaly types will come, so trust the in-game icon, not a published date.
When Hostiles can skip your shields on a roll, a shield-mitigation crew is no longer the automatic best choice. A hull-mitigation crew holds up better. Treat this as one workable answer, not the only one.